How to Multiply a Fraction by a Whole Number

To multiply a fraction by a whole number:

1. Multiply the numerator of the fraction by the whole number.
2. Keep the denominator the same.
3. Simplify the fraction if possible.

For example, multiply 5 ×   ²/₇  .

The numerator of the fraction is the number on top, which is 2.

We multiply 2 by 5 but keep the denominator of 7 as 7.

5 ×   ²/₇   =   ¹⁰/₇.

We simplify if possible. Since the fraction is an improper fraction, we can convert it into a mixed number.

  ¹⁰/₇   means 10 ÷ 7, which is 1 remainder 3. Therefore the fraction can be written as 1   ³/₇.

Alternative Method for Multiplying a Fraction by a Whole Number

For example, multiply 4 ×   ¹/₂.

The first step is to write 4 as   ⁴/₁.

The second step is to multiply the numerators: 4 × 1 = 4. The numerator of the answer is 4.

The third step is to multiply the denominators: 1 × 2 = 2. The denominator of the answer is 2.

Therefore 4 ×   ¹/₂   =   ⁴/₂.

Finally, simplify the fraction by dividing both the numerator and denominator by the same value. We can divide 4 and 2 by 2 so   ⁴/₂   simplifies to   ²/₁.   ²/₁   is the same as 2.

Therefore 4 ×   ¹/₂   = 2.

We also know that one half of 4 is 2.

How to Multiply a Fraction by a Whole Number in Simplest Form

To multiply a fraction by a whole number, multiply the numerator by the whole number. To write this answer in the simplest form, divide the numerator and denominator by the largest number that divides into both exactly.

For example, work out 2 ×   ³/₁₀   in simplest form.

The first step is to multiply the numerator of the fraction by the whole number. 2 × 3 = 6 and so 2 ×   ³/₁₀   =   ⁶/₁₀.

The second step is to simplify the fraction by dividing the numerator and denominator by the largest number that divides into both.

Both 6 and 10 can be divided by 2.   ⁶/₁₀   simplifies to   ³/₅.

Therefore 2 ×   ³/₁₀   in simplest form is   ³/₅.

Multiplying a fraction by a whole number can also be calculated by dividing the denominator by the whole number.

10 is the denominator and 10 ÷ 2 = 5, which is the new denominator. This only works if the denominator of the fraction can be divided by the whole number.

How to Multiply a Mixed Number by a Whole Number

To multiply a mixed number by a whole number:

1. Convert the mixed number into an improper fraction.
2. Multiply the numerator of the improper fraction by the whole number.
3. Simplify if possible and convert back to a mixed number.

For example, multiply 2 × 1   ²/₃.

Step 1. Convert the mixed number into an improper fraction.

Keep the denominator the same.

To find the new numerator, multiply the whole number of the mixed number with the denominator and then add the numerator.

The denominator is 3. The numerator is found by multiplying 1 and 3 to make 3 and then adding 2 to make 5.

1   ²/₃   =   ⁵/₃.

Step 2. Multiply the numerator of the improper fraction by the whole number.

We multiply 2 × 5 = 10 and so 2 ×   ⁵/₃   =   ¹⁰/₃.

The final step is to simplify and write as a mixed number again.

  ¹⁰/₃   = 3   ¹/₃  .

Therefore 2 × 1   ²/₃   = 3   ¹/₃

Multiplying a Fraction by a Whole Number Using a Number Line

Mark the fraction on the number line. To multiply it by a whole number, add on the same fraction as many times as the multiplication requires.

For example here is 5 ×   ¹/₈   on a number line.

We split each whole number into eighths and count five of them.

5 ×   ¹/₈   =   ⁵/₈.

Here is another example involving an improper fraction or mixed number.

Work out 5 ×   ¹/₃   using a number line.

We split each whole number into thirds. We then count five of these jumps on our number line.

5 ×   ¹/₃   =   ⁵/₃.

As a mixed number this is 1   ²/₃.

Multiplying a Fraction by a Whole Number using Models

Models can be used to teach the process of multiplying fractions by whole numbers.

Here is a model of the fraction   ¹/₃. To multiply it by 2, we have twice as many parts.

We can see that we now have   ²/₃   shaded in.

Here is another example of using a model to multiply   ¹/₄   by 3.

When we multiply   ¹/₄   by 3, we now have   ³/₄  .

Models are useful when multiplying fractions and whole numbers because we can see that the denominator of the fraction does not change because the number of parts in the model remains the same.

Multiplying Fractions and Whole Numbers Word Problems

Here are some examples of word problems involving multiplying fractions and whole numbers.

Q1. I cycle   ¹/₃   of a kilometre every day. How many kilometres do I cycle in 5 days?

Answer:   ¹/₃   × 5 =   ⁵/₃.

Q2. At a party, each person needs   ³/₄   of a litre of drink. 10 people are coming in total. How many litres of drink should be bought?

Answer:   ³/₄   × 10 =   ³⁰/₄.

Q3. My phone battery charges up by   ³/₁₀₀   every minute. What fraction will it have charged up after 7 minutes?

Answer:   ³/₁₀₀   × 7 =   ²¹/₁₀₀.

What is Rotational Symmetry?

Rotational symmetry is when an object has been rotated but it looks like it is in its original orientation. For example, a square will look the same when it is rotated a quarter turn. A shape does not have rotational symmetry if it does not look the same when rotated. For example, a kite has no rotational symmetry.

Below is a square showing rotational symmetry. We can see that a square can be rotated to 4 different positions and look the same.

The outside of the shape looks the same in the 4 different positions. When a shape demonstrates rotational symmetry, it looks like it has not been rotated at all.

A kite is an example of a shape with no rotational symmetry. A shape with no rotational symmetry only fits into its original outline once. Shapes with no rotational symmetry have rotational symmetry order 1.

We can see that the only time that the kite matches its outline is when it has not been rotated.

What is the Order of Rotational Symmetry?

The order of rotational symmetry (or degree of rotational symmetry) is the number of times an object looks the same when it is rotated a full turn of 360°. The shape must look as though it has not rotated at all. For example, a rectangle has rotational symmetry of order 2. When it is turned 180° and 360° it looks the same as when it is at its starting position.

We can see that a rectangle always has rotational symmetry of order 2.

Here are two different rectangles with both rotational symmetries shown.

We only count rotational symmetry over one full turn. Once the shape has returned to its original position, we stop. Otherwise we would continue to count onwards forever.

How to Find the Order of Rotational Symmetry

To find the order of rotational symmetry:

1. Rotate the shape one full turn.
2. As it rotates, count the number of times the shape looks the same as before it was rotated.
3. This number is the order of rotational symmetry.

For example, find the order of rotational symmetry for this shape.

Step 1 is to rotate the shape around one full turn.

Step 2 is to count the number of times the shape looks the same as before it was rotated. This is when it fits into its original outline.

The diagram above shows the two rotational symmetries of this shape.

The first symmetry is the position of the original shape and the second symmetry is the shape upside down after a rotation of 180°.

Rotational Symmetry Using Tracing Paper

To find the order of rotational symmetry with tracing paper:

1. Place the tracing paper over the shape.
2. Draw around the outside of the shape.
3. Place your pencil in the centre of the shape and rotate the paper a full turn without moving the shape below.
4. Count how many times the drawing on the paper matches up with the outline of the shape below.
5. This number is the order of rotational symmetry.

For example, use tracing paper to find the order of rotational symmetry for this equilateral triangle.

First place tracing paper over the shape.

Then draw around it.

Then rotate the paper a full turn, counting the number of times that the drawing matches the shape below.

This triangle matches up 3 times and so, the order of rotational symmetry is 3.

Letters of the Alphabet with Rotational Symmetry

Capital letters of the alphabet with rotational symmetry are : H, I, N, O, S, X, Z. These letters all have rotational symmetry order 2 because they look the same after rotating half a turn and a full turn. All of the other capital letters of the alphabet have no rotational symmetry.

The only letters of the alphabet with rotational symmetry are H, I, N, O, S, X and Z. Below we can see the letters of H, I, N, S and Z, which all have rotational symmetry order 2.

Both of the capital letters O and X also have order of rotational symmetry 2.

It may look like X has a rotational symmetry of order 4, however, it is slightly longer than it is wide and it does not look the same when rotated 90 degrees.

The letter O is also longer than it is wide and does not look exactly the same when rotated 90 degrees.

What is the Difference Between Rotational and Reflective Symmetry?

Rotational symmetry is how many times a shape fits into its original image when rotated a full turn. Reflective symmetry (or line symmetry) is the number of lines of symmetry pass through the centre of the shape so that both sides of the line look the same. The order of rotational symmetry is not the necessarily the same as the order of reflective symmetry.

In the triangle below we can see that the rotational symmetry is order 3 because the triangle can fit into its original shape 3 times when rotated. The reflective symmetry is also order 3 because there are 3 mirror lines of symmetry.

In the kite below, the kite only fits into itself once and so, the order of rotational symmetry is 1.

There is one line of reflective symmetry going through the middle. Each side of this line looks identical.

The order of rotational symmetry is not always the same as the order of reflective symmetry. For example in a trapezoid, there are no lines of symmetry and so the order of reflective symmetry is 0. However, the trapezoid fits into its outline once and so the order of rotational symmetry is 1.

The main difference between rotational symmetry and reflective symmetry is that a shape always has at least one degree of rotational symmetry when it fits into its own outline. A shape does not necessarily have any lines of reflective symmetry.

Here is a table showing the difference in rotational and reflective symmetry for several shapes.

What is the Angle of Rotational Symmetry?

The angle of rotational symmetry is the smallest angle that a shape must be rotated to fit within its original outline. When rotated through this angle, it must look as though it has not been rotated at all. The angle of rotational symmetry can be calculated by dividing 360° by the order of rotational symmetry of the shape.

For example, a square has an angle of rotational symmetry of 90°. Rotating a square by 90° results in the square looking identical to before.

The angle of rotational symmetry can also be found by dividing 360° by the order of rotational symmetry of the shape.

A square has a rotational symmetry order 4. This means that there are 4 positions that a square can be rotated into where it looks the same as before it was rotated.

360° ÷ 4 = 90° and so the angle of rotational symmetry is 90°.

Here is a list of the angle of rotational symmetry for various shapes:

Can a Shape have Rotational Symmetry of Order Zero?

No shape can have rotational symmetry of order zero. The smallest order of rotational symmetry a shape can have is 1. This is because all shapes must look the same when rotated a full turn to their original position. A shape with no rotational symmetry has rotational symmetry order 1.

Real Life Examples of Rotational Symmetry

Examples of rotational symmetry in real life include windmills, wheels, fidget spinners, fan blades, ferris wheels and the recycling logo.

Here is an example of a windmill fan showing rotational symmetry of order 3.

Here is an example of a ferris wheel which looks the same whenever a new carriage reaches the top.

Rotational Symmetry in Nature

Examples of rotational symmetry in nature include flower petals, starfish, snowflakes and segments of fruit. These items will all look the same if rotated.

Here is an example of rotational symmetry in nature. This plant has four leaves that all look the same. The order of rotational symmetry is 4.

Here is an example of a snowflake with order of rotational symmetry 6.

A starfish is an example of rotational symmetry found in nature. It has rotational symmetry order 5 due to each of its 5 legs.

Here is a fruit showing rotational symmetry. Rotating the cut piece of fruit, each segment looks the same.

Here is a flower showing rotational symmetry.

Examples of Rotational Symmetry

Here is a list of shapes and their order of rotational symmetry:

Rotational Symmetry of a Square

A square has rotational symmetry order 4. The angle of rotational symmetry is 90° since a square looks the same if it is rotated at 90°, 180°, 270° and 360°.

Rotational Symmetry of a Rectangle

A rectangle has rotational symmetry order 2. The angle of rotational symetry is 180° since a rectangle looks the same if it is rotated at 180° and 360°.

Rotational Symmetry of a Parallelogram

A parallelogram has rotational symmetry order 2. The angle of rotational symetry is 180° since a parallelogram looks the same if it is rotated at 180° and 360°.

Rotational Symmetry of a Rhombus

A rhombus has rotational symmetry order 2. The angle of rotational symetry is 180° since a rhombus looks the same if it is rotated at 180° and 360°.

Rotational Symmetry of a Kite

A kite has no rotational symmetry. It only fits into its shape when it is not rotated. A kite has an order of rotational symmetry of 1.

Rotational Symmetry of a Trapezoid

A trapezoid has no rotational symmetry. It only fits into its shape when it is not rotated. A trapezoid has an order of rotational symmetry of 1.

Rotational Symmetry of a Triangle

An equilateral triangle has a rotational symmetry of order 3. This means that it can fit into itself three times when rotated. All other triangles such as isosceles, scalene and right-angled triangles all have rotational symmetry of order 1.

Here is a table summarising the rotational symmetry of triangles:

Rotational Symmetry of a Pentagon

A regular pentagon has rotational symmetry of order 5. This means that it looks the same in 5 different positions when rotated over a full turn. The angle of rotational symmetry of a regular pentagon is 72°. If a pentagon is not regular, it has no rotational symmetry.

Rotational Symmetry of a Hexagon

A regular hexagon has rotational symmetry of order 6. This means that it looks the same in 6 different positions when rotated over a full turn. The angle of rotational symmetry of a regular hexagon is 60°.

Rotational Symmetry of a Circle

A circle is the only shape that has infinite rotational symmetry. This means that it looks the same no matter what angle it is rotated. No other shape has this property. A circle is also the only shape that contains an infinite number of lines of reflective symmetry.

A circle is the only shape that has rotational symmetry of infinite order and reflective symmetry of infinite order.

Rotational Symmetry of a Star

A star has the same rotational symmetry as the number of points it contains. For example, a 4-pointed star has rotational symmetry order 4 and a 5-pointed star has rotational symmetry order 5.

Here is a 4-pointed star. We can see that the star looks the same every 90 degrees.

Here is a 5-pointed star. It looks the same every rotation of 72 degrees.

Rotational Symmetry of a Regular Polygon

All regular polygons have a rotational symmetry equal to the number of sides. For example, an equilateral triangle has rotational symmetry of order 3, a square has rotational symmetry order 4 and a regular nonagon has rotational symmetry order 9.

A regular polygon is a shape with sides of equal length.

Simply count the number of sides of a regular polygon to find its order of rotational symmetry.

An equilateral triangle has rotational symmetry order 3.

A square has rotational symmetry of order 4.

A regular pentagon has rotational symmetry of order 5.

Shapes with Rotational Symmetry Order 1

If a shape has rotational symmetry order 1, then it has no rotational symmetry. If it is rotated then it will not look the same as its original image.

How to Calculate the Elapsed Time

To calculate elapsed time:

1. Count the number of minutes from the first time to the hour that comes after it.
2. Count on in hours until the hour immediately before the final time.
3. Count the number of minutes from this hour until the final time.
4. Add the hours and minutes to find a total.

For example, find the elapsed time between 11:45 and 15:11.

The first step is to count the number of minutes from the first time to the hour that comes after it. The first time is 11:45 and the hour that comes after it is 12:00.

Each hour has 60 minutes and so, to work out the number of minutes between 11:45 and 12:00, subtract 45 from 60. 60 – 45 = 15 and so, there are 15 minutes between 11:45 and 12:00.

The second step is to count on in hours until the hour immediately before the final time. From 12:00 until 15:00, there are 3 hours.

The third step is to count the number of minutes from this hour until the final time. From 15:00 until 15:11, there are 11 minutes. Simply read off the minutes from the final time.

The final step is to add the hours and minutes to find a total. There are 3 hours in total. Counting the minutes, 15 + 11 = 26.

The elapsed time is 3 hours and 26 minutes.

Examples of Calculating Elapsed Time

For example, calculate the elapsed time between 08:20 and 10:15. First count 40 minutes from 08:20 until the next hour of 09:00. Then count the 1 hour from 09:00 until 10:00. Then count the 15 minutes from 10:00 to 10:15. Adding the hours and minutes separately, the total elapsed time is 1 hour 15 minutes.

Here is a set of examples of calculating the elapsed time between two given times.

Elapsed Time Using Subtraction

If the number of minutes in the final time is larger than the number of minutes in the start time, the elapsed time can be calculated by subtracting the start time from the final time. Subtract the hours and minutes separately. For example, between 10:22 and 23:57, we subtract the hours: 23 – 10 = 13. Then subtract the minutes: 57 – 22 = 35. The elapsed time is 13 hours and 35 minutes.

The elapsed time can be calculated by direct subtraction only if the number of minutes in the final time is greater than the number of minutes in the start time. If this is not the case, negative numbers will arise if a larger number is subtracted from a smaller number.

The Formula for Calculating Elapsed Time

The formula for calculating elapsed time is elapsed time = end time – start time. Subtract the minutes and hours separately. For example to calculate the elapsed time between 12:10 and 16:40, subtract 12:10 from 16:4. Looking at the hours, 16 – 12 = 4 and looking at the minutes, 40 – 10 = 30. The elapsed time is 4 hours and 30 minutes.

Elapsed Time on a Number Line

To work out elapsed time on a number line, write the start time on the left of the line and the finish time on the right of the line. Write all of the hours in between these times on the number line. Work out the minutes from the start time to the first hour and add this to the number of minutes from the last hour until the finish time. Count the number of whole hours in between.

For example, use a number line to find the elapsed time between 10:50 and 11:43.

First count on 10 minutes to get to 11:00 and then count on 43 minutes to get to 11:43.

In total, the elapsed time is 53 minutes.

Elapsed Time Word Problems

Here are some elapsed time worded problems.

To solve word problems with elapsed time, identify the start and finish times. Write these times at each end of a number line. Find the minutes from the start time to the next hour and from the final hour until the finish time. Count the hours in between.

For example, “I start my homework at 18:30 and finish it at 19:13. How long did it take to complete it?”

There are 30 minutes from the start time of 18:30 to the next hour and there are 13 minutes from 19:00 until 19:13.

The total minutes spent on the homework is 30 + 13, which equals 43 minutes.

Here is another word problem involving calculating elapsed time.

“The bus leaves the station at 07:50 and arrives at its destination at 11:23. How long did the journey take?”

There are 10 minutes until the next hour of 08:00 and there are 23 minutes from 11:00 until 11:23.

In total, 10 + 23 = 33 minutes.

There are also 3 hours between 08:00 and 11:00.

The total elapsed time is 3 hours and 33 minutes.

How to Find the Area of a Trapezoid

To find the area of a trapezoid:

1. Add the two parallel side lengths.
2. Multiply this by the distance between the parallel sides.
3. Divide this by 2.

For example, find the area of the trapezoid with parallel side lengths 6m and 10m, with a distance of 4 m between them.

1. Add the parallel sides: 6 + 10 = 16.
2. Multiply this by the distance between them: 16 × 4 = 64.
3. Divide by 2: 64 ÷ 2 = 32.

And so the area is 32 m².

Alternatively, the area of a trapezoid can be found by using the formula Area = ¹/₂ (a+b)h, where a and b are the parallel sides of the trapezoid and h is the distance between them.

a = 6 and b = 10, while h = 4.

Area = ¹/₂ (a+b)h becomes Area = ¹/₂ × (6 + 10) × 4.

This becomes Area = ¹/₂ × 16 × 4, which equals 32 m².

Formula for the Area of a Trapezoid

The formula for the area of a trapezoid is Area = ^h/₂ (a+b), where a and b are the two parallel side lengths and h is the distance between them. For example, if a trapezoid has parallel sides of 5m and 7m and a distance between them of 4m, then a = 5, b = 7 and h = 4. The area formula becomes ⁴/₂ (5+7) which equals 24 m².

It does not matter which of the two parallel sides are labelled as a and b.

The formula for the area of a trapezoid can also be written as Area = ^(a+b)/₂ h.

It can also be written as Area = ¹/₂ h(a+b) or Area = ¹/₂ (a+b)h.

All of these formula are just rearrangements of the same formula and will give the same answer.

Area of a Right-Angled Trapezium

To find the area of a right-angled trapezoid, use the formula Area = ¹/₂ h(a+b), where a and b are the two parallel sides and h is the distance between them.

For example, calculate the area of the following right trapezoid.

The two parallel sides are 5 m and 9 m, so a = 5 and b = 9. It does not matter which side is a or which side is b.

The distance between the parallel sides is 2 m and so, h = 2.

To find the area, substitue the values of h = 2, a = 5 and b = 9 into Area = ¹/₂ h(a+b).

Area = ¹/₂ h(a+b) becomes Area = ¹/₂ × 2 × (5 + 9)

This becomes Area = ¹/₂ × 2 × 14 which equals 14 m².

Area of an Isosceles Trapezoid

An isosceles trapezoid has diagonal sides that are the same length. These diagonal side lengths do not affect the area. To find the area of an isosceles trapezoid, use the formula Area = ¹/₂ h(a+b), where a and b are the lengths of the two parallel sides and h is the distance between them.

For example, here is an isosceles trapezoid. Its diagonal sides are the same size and it is symmetrical about its middle.

The two parallel side lengths are 4 m and 8 m. So a = 4 and b = 8. It does not matter which side is a and which side is b out of these two values.

The distance between the parallel sides is 3 m. So h = 3.

We substitute the values of h = 3, a = 4 and b = 8 into the formula Area = ¹/₂ h(a+b).

It becomes Area = ¹/₂ × 3 × (4 + 8), which becomes Area = ¹/₂ × 3 × 12.

It is easier to halve 12 than 3 so we halve the 12 to get 6.

Area = 3 × 6, which equals 18 cm².

The area is measured in cm² because the sides of the trapezoid are measured in cm.

Area of an Irregular Trapezoid

The area of an irregular trapezoid is found in the usual manner using Area = ¹/₂ h(a+b). a and b are the two parallel sides and h is the distance between them. It does not matter what length the diagonal sides are in an irregular trapezoid as they do not affect the area.

For example, find the area of the irregular trapezoid shown.

The two parallel sides are 6 m and 10 m. So a = 6 and b = 10. It does not matter which sides are a and b as long as they are parallel.

The distance between the parallel sides is 4 m, so h = 4.

The area of the irregular trapezoid is found using the formula Area = ¹/₂ h(a+b).

Substituting in h = 4, a = 6 and b = 10, we get Area = ¹/₂ × 4 × (6 + 10).

This becomes Area = ¹/₂ × 4 × 16. Half of 16 is 8 and 8 × 4 = 32. The area is 32 m².

Area of a Trapezoid without the Height

If the height of a trapezoid is not given, it must be calculated using Pythagoras’ theorem. To do this, imagine a right-angled triangle formed on one side of the trapezoid, where the diagonal side is the hypotenuse. Square the hypotenuse, subtract the base of the triangle squared and then square root this to find the height of the trapezoid. The area can then be found using A = ¹/₂ h(a+b).

Here is an example of calculating the area of a trapezoid without knowing the height.

The first step is to create a right-angled triangle on one side of the trapezoid. The hypotenuse of the triangle is the diagonal side of the trapezoid, in this case, it is 5 m.

Next, find the base of the triangle, which is the difference between the top and bottom side lengths of the trapezoid. 7 – 3 = 4 and so, the base of the triangle is 4 m.

The next step is to use Pythagoras’ theorem to work out the height of the triangle. This will be the same as the height of the trapezoid.

Since the hypotenuse is known, the height of the triangle is one of the shorter sides of the triangle. To find the height of the triangle, use the formula: $$h = \sqrt{c^2 – b^2}$$ where h is the height, c is the diagonal side of the triangle and b is the base of the triangle.

5² – 4² = 9 and then the square root of 9 is 3. The height is 3.

Now the area of the trapezoid can be found using Area = ¹/₂ h(a+b).

h is the height of the trapezoid, b is the base of the trapezoid and a is the top side of the trapezoid.

Area = ¹/₂ × 3 × (3 + 7) = 15².

Proof of the Area of a Trapezoid Formula

A trapezoid of height ‘h’, and parallel side lengths ‘a’ and ‘b’ can be made out of a triangle and a rectangle combined. The area of the rectangle is ah and the area of the triangle is ¹/₂ h(b-a). Adding the area of the triangle and rectangle together proves the formula for the area of a trapezoid, Area = ¹/₂ h(a+b).

The proof of the area of a trapezoid is shown below.

The trapezoid has height ‘h’, base ‘b’ and a top length of ‘a’. It can be split into two separate shapes, a rectangle and a triangle.

The rectangle area is length × width, which is a × h or ah. We call this Area 1, A₁.

The triangle area is ¹/₂ × base × height. The base length of the triangle is the longer side of the trapezoid subtract the shorter side, b-a. The area of the triangle, A₂ = ¹/₂ h(b-a).

We add the area of the rectangle to the area of the triangle to find the area of the trapezoid.

Area = A₁ + A₂.

Area = ah + ¹/₂ h(b-a).

Expanding the second term we get:

Area = ah + ¹/₂ bh – ¹/₂ ah.

Subtracting ¹/₂ ah from ah, we get:

Area = ¹/₂ ah + ¹/₂ bh.

Factorising the ¹/₂ and the h, we get the final area of a trapezoid:

Area = ¹/₂ h(a + b).

Here is an alternative proof of the area of a trapezoid in which we put two trapezoids together to form a rectangle.

Two identical trapezoids with height ‘h’, base ‘b’ and top side ‘a’ can be placed together to form a rectangle. This rectangle has height ‘h’ and a base of length ‘a+b’. The area of the rectangle is base × height, which is h × (a+b). The area of each individualy trapezoid is half of this, which is ¹/₂ h(a+b).

The area of a rectangle is base × height. We can see that the height of the rectangle is the same as the height of each trapezium and the base of the rectangle is made up of the base of the trapezium and the top side of the trapezium combined. The base of the rectangle is a+b.

The area of the rectangle is h(a+b), however, each trapezoid is half of the total rectangle area because there are two of them. We halve the area of the rectangle to prove the area of the trapezoid.

Area = ¹/₂ h(a+b).

What Does Simplifying a Ratio Mean?

Simplifying a ratio means dividing all of the numbers in a ratio by the same number to make smaller numbers. All of the numbers in the ratio must remain as whole numbers. If there is no number larger than one that divides exactly into all of the numbers in the ratio, then the ratio is fully simplified.

Here are 9 rectangles and 3 triangles. We say that the rectangles and triangles are in a ratio of 9:3. This ratio is pronounced as ‘9 to 3’.

Here we can see that the ratio of 9:3 can be simplified to 3:1 by dividing both numbers by 3.

The rectangles and triangles are in a ratio of 3:1, which means that there are 3 rectangles for every 1 triangle.

We can see that in each group, there are three rectangles to one triangle.

Because there is no number larger than one that divides into both 3 and 1, the ratio of 3:1 is fully simplified.

How to Simplify a Ratio

To simplify a ratio, divide each number in the ratio by the same amount. To fully simplify a ratio, divide each number in the ratio by their highest common factor. Always keep the numbers in the ratio as whole numbers. For example, the ratio 9:3 simplifies to 3:1 by dividing by 3.

Factors are numbers that divide exactly into other numbers. It can help to list the factors of each number in a ratio before simplifying it.

For example, simplify the ratio 21:15.

We can list factors of 21 which are: 1, 3, 7 and 21. All of these numbers divide exactly into 21.

We can list factors of 15 which are: 1, 3, 5 and 15. All of these numbers divide exactly into 15.

We can see that the number 3 divides into both 21 and 15 and so, this is the highest common factor. We will divide both numbers in the ratio by 3 to simplify it.

21:15 simplifies to 7:5.

How to Simplify a Ratio with 3 Numbers

To simplify a ratio with 3 numbers, divide all 3 of the numbers in the ratio by the same amount until they cannot be divided exactly any further.

For example, simplify the ratio with the 3 numbers 20:5:10.

Each of 20, 5 and 10 can be divided exactly by 5.

The ratio of 20:5:10 simplifies to 4:1:2.

It can help to list the factors of all 3 numbers before simplifying. Then to simplify in one go, divide all of the numbers by the largest number that is a factor of all of the numbers. This is called dividing by the highest common factor.

Here is another example of simplifying a ratio with 3 numbers.

Simplify the ratio of 8:12:4.

The largest number that divides into each of 8, 12 and 4 is 4.

Dividing each of the numbers in 8:12:4 by 4, the ratio is simplified to 2:3:1.

How to Simplify Ratios with Large Numbers

Large ratios can be simplified in steps. First, list the factors of each number. Then divide all of the numbers in the ratio by a number that is a factor of all of the numbers in the ratio. Keep repeating these steps until the numbers cannot be divided exactly anymore.

For example, simplify the ratio 200:150.

The ratio will be simplified in steps.

We can see that both 200 and 150 end in a zero and so, they can both be divided exactly by 10.

This simplifies the ratio 200:150 to 20:15.

The ratio 20:15 can be simplified further by dividing by 5.

20:15 simplifies to 4:3.

200:150 is fully simplified as 4:3.

It is possible to simplify 200:150 to 4:3 in one step by dividing both numbers by 50. However, it can be easier to simplify the numbers in steps.

How to Simplify a Ratio Involving Fractions

To simplify a ratio involving fractions:

1. Rewrite the fractions with equivalent fractions so that they all have the same denominator.
2. Write the numerators of these fractions as a ratio.
3. Simplify this ratio if possible by dividing by the highest common factor.

Simplify the ratio of the fractions ¹/₂ : ²/₃.

¹/₂ and ²/₃ have denominators of 2 and 3 respectively. Both fractions can be made into equivalent fractions with a denominator of 6.

¹/₂ can be written as ³/₆ and ²/₃ can be written as ⁴/₆.

The next step is to write the numerators of the fractions as a ratio. Technically, both fractions are multiplied by 6 to remove the denominator.

The numerators are now 3 and 4 respectively and so, the ratio is 3:4.

There is a shortcut to simplifying ratios containing 2 fractions.

We can multiply the denominator of one fraction by the numerator of the other fraction.

We multiply the denominator of ²/₃ by the numerator of ¹/₂.

We multiply the denominator of ¹/₂ by the numerator of ²/₃.

3 × 1 = 3 and 2 × 2 = 4.

The ratio is 3:4.

How to Simplify a Ratio Involving Decimals

To simplify a ratio with decimals:

1. Multiply all of the numbers in the ratio by one of 10, 100, 1000 etc. so that the decimals are all whole numbers.
2. Simplify this ratio by dividing the numbers in the ratio by their highest common factor.

For example, simplify the ratio containing the decimals 0.2:1.6.

Multiplying both decimals by 10 turns the ratio into 2:16. The ratio is now written with whole numbers.

Now the ratio can be simplified.

Dividing both numbers by 2, the ratio 2:16 simplifies to 1:8.

A ratio can contain decimal numbers. However, when a ratio is simplified, the decimal numbers are multiplied to make whole numbers. A simplified ratio should not contain decimal numbers.

Here is another example of simplifying decimal ratios.

Simplify the ratio of 0.5:0.15.

Since 0.15 contains 2 decimal places, we multiply it by 100 to make it a whole number.

Multiplying both 0.5 and 0.15 by 100, the ratio becomes 50:15.

This ratio can be fully simplified by dividing both numbers by 5.

50:15 simplifies to 10:3.

How to Simplify a Ratio with Different Units

To simplify ratios with different units:

1. Convert the larger unit to the smaller unit.
2. Simplify the ratio by dividing the numbers by their highest common factor

The numbers in a ratio must be converted to have the same units when they are simplified.

For example, simplify the ratio of 2 cm : 15 mm. We can see that one unit is centimetres and the other is millimetres.

The first step is to convert the larger unit (centimetres) to the smaller unit (millimetres). We do this by multiplying by 10, since there are 10 mm in 1 cm.

2 cm: 15 mm becomes 20 mm : 15 mm.

This ratio can be simplified by dividing by 5.

20 mm: 15 mm becomes 4:3. Because the units are both the same (mm), we do not need to write them in our answer.

Here is another example of simplifying a ratio with different units.

Simplify 65 cm: 1m.

We first convert metres to centimetres by multiplying by 100. There are 100 cm in 1 m.

65 cm: 1m becomes 65 cm : 100 cm.

We can now simplify by dividing by 5.

65 cm : 100 cm simplifies to 13:20.

Again we do not need to write the units because they are the same.

Why Simplify a Ratio

Ratios are simplified to find smaller numbers that are easier to work with. Simplifying ratios may also remove any fractions or decimals that may be there. This results in a ratio that is easier to understand and do calculations with.

How to Calculate the Area of a Triangle with 3 Known Sides

To calculate the area of a triangle with 3 known sides, use Heron’s Formula. Heron’s formula is Area = √( s(s-a)(s-b)(s-c) ), where a, b and c are the 3 side lengths of the triangle and s = (a + b + c) ÷ 2. Simply find the values of s, a, b and c and substitute these into the formula for the area.

The steps to find the area of a triangle with 3 sides (a, b and c) are:

1. Work out s = (a + b + c) ÷ 2.
2. Substitute the values of s, a, b and c into the formula of Area = √( s(s-a)(s-b)(s-c) ).

For example, find the area of a triangle with side lengths of 8 m, 3 m and 9 m.

It does not matter which sides are a, b or c.

We will set a = 8, b = 3 and c = 9.

The first step is to work out the semi-perimeter, s. The semi-perimeter is simply half of the perimeter. We find the semi-perimeter by adding up the side lengths and dividing by 2.

8 + 3 + 9 = 20 and 20 ÷ 2 = 10. Therefore the semi-perimeter is 10.

s = 10.

The second step is to substitute the values of s = 10, a = 8, b = 3 and c = 9 into Heron’s Formula.

Area = √( s(s-a)(s-b)(s-c) ) becomes Area = √( 10(10-8)(10-3)(10-9) ).

This becomes Area = √(10 × 2 × 7 × 1), which simplifies to Area = √140.

Finally, the square root of 140 is calculated using a calculator. Area = 11.8 m².

What is Heron’s Formula

Heron’s formula is Area = √( s(s-a)(s-b)(s-c) ), where a, b and c are the three side lengths of a triangle and s = (a + b + c) ÷ 2. It can be used to calculate the area of any triangle as long as all three side lengths are known. The formula is named after Heron of Alexandria (10 – 70 AD) who discovered it.

Why use Heron’s Formula

Heron’s formula allows us to calculate the area of a triangle as long as all 3 of its sides are known. The advantage of Heron’s formula is that no other lengths or angles of the triangle need to be known. As long as the three side lengths are known, Heron’s formula works for all triangles.

Heron’s Formula for an Isosceles Triangle

Heron’s formula for any triangle is Area = √( s(s-a)(s-b)(s-c) ). For an isosceles triangle, two sides are the same length and we can say that side c = side a. Heron’s formula for an isosceles triangle then becomes Area = √( s(s-a)²(s-b) ), where a is the length of the two equal sides, b is the length of the other side and s = (2a + b) ÷ 2.

For example, here is Heron’s formula for an isosceles triangle with side lengths of 2 cm, 6 cm and 6 cm.

We can use the usual form of Heron’s formula to find the area. The semi-perimeter is the sum of the sides divided by 2.

2 + 6 + 6 = 14 and 14 ÷ 2 = 7. Therefore s = 7.

We can use Area = √s(s-a)(s-b)(s-c), which becomes Area = √7(7-2)(7-6)(7-6). This becomes Area = √35, which equals 5.92 cm².

Here is an example of using the isosceles version of Heron’s formula: Area = √s(s-a)²(s-b). Here a = 6 and b = 2. The semi-perimeter is still 7.

Area = √s(s-a)²(s-b) becomes Area = √7(7-6)²(7-2).

This becomes Area = √35, which equals 5.92 cm². This is the same answer as before and either method can be used.

How to Calculate the Area of a Triangle with 3 Equal Sides

The area of a triangle with 3 equal sides can be calculated with the formula Area = ^√3/₄ a², where a is the length of one of the sides. Alternatively, Heron’s formula for an equilateral triangle is Area = √(s(s-a)³), where a is the side length and s = ^3a/₂.

An equilateral triangle is a triangle with 3 equal side lengths. We will first look at finding the area of an equilateral triangle using Heron’s formula.

Heron’s formula for the area of an equilateral triangle is Area = √(s(s-a)³), where a is the side length.

In the example below, a = 8 metres.

For a triangle that has 3 equal sides, the semi-perimeter is simply s = ^3a/₂. Here with a = 8, s = 12.

Area = √(s(s-a)³) becomes Area = √(12(12-8)³), which becomes Area = √768.

The area is 27.7 m².

Here is another method for calculating the area of an equilateral triangle.

The formula for the area of an equilateral triangle is Area = ^√3/₄ a², where a is the length of one of the sides.

In this example, a = 8 metres.

Area = ^√3/₄ a² becomes Area = ^√3/₄ × 8².

This works out as 27.7 m².

Both methods give the same answer.

What is the Nth Term?

The nth term is a formula used to generate any term of a sequence. To find a given term, substitute the corresponding value of n into the nth term formula. For example, if the nth term is 3n + 2, the 10th term of the sequence can be found by substituting n = 10 into the nth term. 3 × 10 + 2 = 32 and so, the tenth term of the sequence is 32.

The nth term can be used to generate a sequence. To generate a sequence using the nth term, substitute consecutive values of n into the sequence starting from n = 1.

Here is a list of calculations for generating the first 5 terms of the sequence 3n + 2.

• When n = 1, 3 × 1 + 2 = 5. The 1st term is 5.
• When n = 2, 3 × 2 + 2 = 8. The 2nd term is 8.
• When n = 3, 3 × 3 + 2 = 11. The 3rd term is 11.
• When n = 4, 3 × 4 + 2 = 14. The 4th term is 14.
• When n = 5, 3 × 5 + 2 = 17. The 5th term is 17.

Here is a table showing the first ten terms of the sequence 3n + 2 using the nth term formula.

What is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers that always increase or decrease by the same amount from one number to the next. We say that in an arithmetic sequence, there is a common difference between the terms. For example, the sequence of 3, 5, 7, 9, 11… is arithmetic because the difference between each term is 2. We add 2 from one term to the next and so, the common difference is 2.

The number in front of the n in the arithmetic sequence formula tells us what the difference is between each term in the sequence.

For example in the sequence above, the nth term is 2n + 1. The terms increase by 2 each time because there is a 2 in front of the n in the formula.

The most basic arithmetic sequences are the times tables. We can form sequences for the times tables by writing the appropriate number in front of n.

For example, the nth term for the 5 times table is just 5n.

Substituting n = 1, n =2 etc into the 5n sequence we get 5, 10, 15, 20 and so on.

We can see that the common difference between the terms in the 5 times table is 5. We add 5 to get from one number to the next.

The same can be seen in the 3 times table with the 3n sequence.

Again, the 3 times table goes up in threes and is formed by writing a 3 in front of the n.

Arithmetic sequences can also decrease from one term to the next. For example, the arithmetic sequence formed by -2n + 10 starts at 8 and goes down by 2 each time. This is because the number in front of the n is -2.

Decreasing arithmetic sequences always decrease by the same amount from one term to the next. The number in front of the n in the nth term for a decreasing arithmetic sequence tells us how much the terms decrease by each time. For example, with -2n + 10, the terms decrease by 2 each time.

How to Find the Nth Term of an Arithmetic Sequence

To find the nth term of an arithmetic sequence, find the common difference between each term. Start by writing this difference multiplied by n. Then list multiples of the difference and see what needs to be added or subtracted to this to make the original sequence. For example 3, 7, 11, 15, 19… increase by 4 each time so we start with the 4n sequence: 4, 8, 12, 16, 20. We can see that we subtract 1 from each term in the 4n sequence to make the original sequence and so, the nth term is 4n – 1.

The number in front of the n in the nth term formula tells us how much we are increasing or decreasing by.

Since the terms in this example increase by 4, we write a 4 in front of the n.

The 4n sequence is the four times table. We compare each term in the 4 times table to see what we need to adjust it by to make our sequence. It helps to list the two sequences above each other to compare them.

If we subtract 1 from the 4n sequence, we get the numbers in our original sequence. For example, 4 – 1 = 3, 8 – 1 = 7 and so on.

The nth term is 4n – 1.

How to Easily Find the Nth Term of a Sequence

To find the nth term of a sequence, use these steps:

1. Find the common difference from one term to the next.
2. Write this difference with an ‘n’ after it.
3. Subtract the difference from the first term in the sequence.
4. Add this to the answer written in step 2.

For example, we will find the nth term for the sequence starting 3, 7, 11, 15, 19…

Step 1. Find the common difference: The terms increase by 4 each time.

Step 2. Write this difference with an ‘n’ after it: ‘4n’.

Step 3. Subtract the difference from the 1st term: 3 – 4 = -1.

Step 4. Write this after the answer to step 2: ‘4n – 1’.

And so the nth term of our sequence is 4n – 1.

Nth Term of a Decreasing Sequence

If a sequence is decreasing from one term to the next, the number in front of n in the nth term will have a negative sign. For example, in the sequence 8, 6, 4, 2, 0… the terms decrease by 2 so we start by writing -2n. The -2n sequence is -2, -4, -6, -8, -10 and so on. We need to add 10 to each of these terms to make the original sequence, so the nth term is -2n + 10.

The -2n sequence is simply the negative 2 times table.

We need to add 10 to each of these terms to make the sequence 8, 6, 4, 2, 0…

For instance, -2 + 10 = 8, -4 + 10 = 6 and so on.

Here is another example of finding the nth term of a decreasing sequence. We have -5, -8, -11, -14, -17…

The sequence decreases by 3 from one term to the next. Therefore the common difference is -3.

We start by looking at the -3n sequence, which is just the negative 3 times table: -3, -6, -9, -12, -15…

We need to take away 2 more from the -3n sequence to make our original sequence. For example, -3 – 2 = -5 and -6 – 2 = -8.

Therefore the nth term is -3n – 2.

How to Find the Nth Term of a Sequence with Decimals

The rule for finding the nth term of a sequence still applies to decimal numbers. The difference between each term is written in front of the n and the adjustment that needs to be added is written at the end.

For example, find the nth term for the decimal sequence 0.3, 0.4, 0.5, 0.6…

The difference from one term to the next is 0.1. Therefore we start with 0.1n.

The 0.1n sequence is 0.1, 0.2, 0.3 and so on.

We need to add 0.2 to each of these terms to make our sequence of 0.3, 0.5, 0.6 and so on.

Therefore the nth term of this decimal sequence is 0.1n + 0.2.

Here is another example of finding the nth term of a decimal sequence. We have 1.2, 1.5, 1.8, 2.1 and so on.

The term-to-term difference is 0.3. We add 0.3 from one term to the next.

So we start with the 0.3n sequence: 0.3, 0.6, 0.9 and so on.

We must add 0.9 to the 0.3n sequence to make the original sequence of 1.2, 1.5, 1.8 and so on.

Therefore the nth term of this decimal sequence is 0.3n + 0.9.

Nth Term of an Arithmetic Sequence Examples

Here are several examples of finding the nth term of an arithmetic sequence.

Formula for the Nth Term of an Arithmetic Sequence

The formula to find the nth term of an arithmetic sequence is a_n = a₁ + (n-1)d, where a_n is the nth term, a₁ is the 1st term, n is the term number and d is the common difference. To find the formula for the nth term, we need a₁ and d. For example, in the sequence 5, 7, 9, 11, 13… a₁ = 5 and d = 2. a_n = a₁ + (n-1)d becomes a_n = 5 + 2(n-1), which simplifies to a_n = 2n + 3.

To find the formula for the nth term of an arithmetic sequence, we need to know the difference, ‘d’, and the first term, ‘a₁‘.

In this example, the first term is 5 and the common difference is 2.

Substituting the values of a₁= 5 and d = 2 into a_n = a₁ + (n-1)d, we get a_n = 5 + (n-1)2.

We simplify this by first expanding the brackets by multiplying (n-1) by 2.

We get a_n = 5 + 2n – 2. This simplifies to a_n = 2n + 3.

What is the Multiplicative Inverse?

The multiplicative inverse of a number is what you multiply it by to make 1. For example, the multiplicative inverse of 2 is ¹/₂ because 2 × ¹/₂ = 1.

Half of 2 is 1 and so, the multiplicative inverse of 2 is 1.

How to Find the Multiplicative Inverse

To find the multiplicative inverse of a number, write its reciprocal. For a fraction, this means switching the numerator and denominator. For a whole number, the multiplicative inverse is 1 over this number. For example, the multiplicative inverse of ²/₃ is ³/₂ and the multiplicative inverse of 3 is ¹/₃.

The multiplicative inverse of a number is the value that we multiply it by to make an answer of 1. When we multiply a number by its multiplicative inverse, the answer must be equal to 1.

To find the reciprocal of a fraction, switch the numerator on top with the denominator on the bottom. The fraction will be upside down.

For example, the reciprocal of ²/₃ is ³/₂.

We can see that when we multiply these fractions, the answer is 1 and so, they are multiplicative inverses of each other.

Multipling the numerators we get 6 and multiplying the denominators we get 6.

6 out of 6 is one whole.

In this example, one third of 3 is 1 and so, 3 and ¹/₃ are multiplicative inverses of each other.

Here are some examples of finding the multiplicative inverse.

How to Find the Multiplicative Inverse of a Fraction

The multiplicative inverse of a fraction is its reciprocal. This means that the numerator and denominator values are switched. The multiplicative inverse of the fraction ^a/_b is ^b/_a. For example, the multiplicative inverse of ⁵/₇ is ⁷/₅.

We can see that we simply switch the numerator and denominator of a fraction to make the multiplicative inverse.

It does not matter how large the numerator or denominator of a fraction is. The multiplicative inverse of the fraction is simply the reciprocal of the fraction. Switch the numerator and the denominator.

For example, the multiplicative inverse of ⁹/₂ is ²/₉.

The multiplicative inverse of a proper fraction is an improper fraction and the multiplicative inverse of an improper fraction is a proper fraction.

Multiplicative Inverse of a Unit Fraction

The multiplicative inverse of a unit fraction is the denominator of the unit fraction. For any unit fraction ¹/_n, the multiplicative inverse is n. For example, the multiplicative inverse of ¹/₅ is 5 because ¹/₅ × 5 = 1.

A unit fraction must have a denominator of 1. The rule for finding the multiplicative inverse of a unit fraction is true for all unit fractions.

We can see an example of finding the multiplicative inverse of a unit fraction below. One fifth of 5 is 1.

How to Find the Multiplicative Inverse of a Whole Number

The multiplicative inverse of a whole number is a unit fraction equal to 1 over that number. For any whole number ‘n’, the multiplicative inverse is ¹/_n. For example, the multiplicative inverse of 2 is ¹/₂ because 2 × ¹/₂ = 1.

This rule for the multiplicative inverse of a whole number works for any positive whole number value.

The multiplicative inverse of 2 is ¹/₂ because one half of 2 is 1.

To find the multiplicative inverse of a whole number, write a fraction with this number as the denominator and a numerator of 1.

For example, the multiplicative inverse of 3 is ¹/₃. The numerator is 1 and the denominator is 3.

The multiplicative inverse of 7 is ¹/₇. This is because 7 × ¹/₇ = 1.

The multiplicative inverse of 13 is ¹/₁₃. This is because 13 × ¹/₁₃ = 1.

Multiplicative Inverse of a Negative Number

The multiplicative inverse of a negative number is also negative. This is because the two negatives must cancel out when multiplied to make 1. For example, the multiplicative inverse of –²/₅ is –⁵/₂ because –²/₅ × –⁵/₂ = 1.

The multiplicative inverse of a number is the value we multiply it by to make an answer of 1.

For a fraction, the multiplicative inverse is simply the reciprical of the fraction. Below is an example of a negative fraction.

We switch the numerator and denominator to find the reciprocal.

The multiplicative inverse of a negative number must also be negative. This is because two negatives multiplied together make a positive and when a number is multiplied by its multiplicative inverse, the answer must be positive 1.

Here is another example. We will find the multiplicative inverse of a negative whole number.

The multiplicative inverse of 3 is ¹/₃. This is because one third of 3 is 1.

The multiplicative inverse of -3 is –¹/₃. This is because a negative number must be multiplied by another negative number to make a positive answer.

What is the Multiplicative Inverse of 1?

The multiplicative inverse of a number is the value you multiply it by to make an answer of 1. Since 1 × 1 = 1, the multiplicative inverse of 1 is 1 itself.

The only number that 1 can be multiplied by to make an answer of 1 is 1.

1 is the multiplicative inverse of 1.

1 can be written as a fraction as ¹/₁. To find the reciprocal of this fraction, we flip it so that the numerator becomes the denominator and vice versa.

The reciprocal of ¹/₁ is ¹/₁, which is the same as 1.

The reciprocal of 1 is 1.

What is the Multiplicative Inverse of Zero?

There is no multiplicative inverse of zero. This is because there is no number that zero can be multiplied by to make 1. Zero is the only number that does not have a multiplicative inverse.

How to Find the Volume of a Pyramid

To find the volume of a pyramid, multiply the base area by the height and divide this by three. The volume of a pyramid is found with the formula V = B × h ÷ 3, where B is the base area and h is the height of the pyramid.

The base area depends on the shape of the base. No matter what the base, the volume of a pyramid is found by multiplying the area of the base by the height of the pyramid and then dividing by 3.

For example for a triangular-based pyramid with a base area of 10 cm² and a height of 6 cm, the volume is found by multiplying the area by the height and dividing by 3. 10 × 6 ÷ 3 = 20 and so, the volume of the pyramid is 20 cm³.

The base is a triangle and the area of a triangle is ¹/₂ × base × height. The base of the triangle is 4 cm and the height of the triangle is 5 cm. The height of the triangle is at right angles to the base of the triangle and it is different to the height of the pyramid, which is 6 cm.

Area = ¹/₂ × base × height and so Area = ¹/₂ × 4 × 5, which equals 10 cm².

Once the base area has been found, multiply this by the height of the pyramid and divide by 3 to find the volume.

10 × 6 ÷ 3 = 20 and so, the volume is 20 cm³.

Volume of a Rectangular-Based Pyramid

To find the volume of a rectangular-based pyramid, multiply the area of the rectangular base by the height of the pyramid and divide by 3. The volume of a rectangular-based pyramid is found by the formula V = lwh ÷ 3, where l is the length of the rectangle, w is the width of the rectangle and h is the height of the pyramid.

The area of a rectangle is length × width. The area of the rectangular base is multiplied by the height of the pyramid and then divided by 3.

The formula for the volume of a rectangular-based pyramid can be written as V = ^{l × w × h}/₃ or as V = l × w × h ÷ 3.

For example, here is a rectangular pyramid with base length of 10 cm, base width of 2 cm and a height of 6 cm.

The volume is found by multiplying the area of the base by the height of the pyramid and dividing by 3.

The area of a rectangle is length × width, which in this example is 10 × 2. The area of the rectangular base is 20 cm^2/.

To find the volume, multiply this area by the height and divide by 3.

20 × 6 ÷ 3 = 40 and so, the volume of the pyramid is 40 cm³.

Volume of a Square-Based Pyramid

To find the volume of a square-based pyramid, multiply the area of the square base by the height of the pyramid and divide this by 3. The volume of a square-based pyramid is found with the formula V = l² × h ÷ 3, where l is the length of the side of the square base and h is the height of the pyramid.

The side lengths of a square are all the same size and so, the area of the square base is simply one side multiplied by itself. For a square with side length l, the area of the square is l².

For example in a square-based pyramid with a side length of 2 cm and a height of 3 cm, the volume is found by multiplying the area of the square base by the height of the pyramid and dividing by 3.

The area of the square base is the length squared. We simply multiply the length of one side of the square by itself.

2 × 2 = 4 and so, the area of the base is 4 cm².

To find the volume, we multiply the area of this base by the height and divide by 3.

The height is 3, so the volume is 4 × 3 ÷ 3. The volume is 4 cm³.

Volume of a Triangular-Based Pyramid

To find the volume of a triangular-based pyramid, multiply the area of the trianglular base by the height of the pyramid and divide this by 3. The volume of a triangular-based pyramid is found with the formula: Volume = ¹/₃ × Base Area × Height, where the base area is ¹/₂ × Triangle Base × Triangle Height.

The base area of a triangular pyramid is a triangle and the area of a triangle is ¹/₂ × Triangle Base × Triangle Height.

In this example, we will call the height of the triangular base ‘h’ and the height of the pyramid ‘H’.

V = ¹/₂ × b × h × H ÷ 3.

For example, in a triangular pyramid with triangle base of 4 cm, triangle height of 5 cm and pyramid height of 6 cm, the volume is found by V = ¹/₂ × 4 × 5 × 6 ÷ 3. The volume is 20 cm³.

The volume of a triangular-based pyramid can be written most simply as V = bhH ÷ 6, where b is the base of the triangle, h is the height of the triangle and H is the height of the pyramid.

In this example, b = 4, h = 5 and H = 6.

V = bhH ÷ 6 and so, V = 4 × 5 × 6 ÷ 6.

V = 20 cm³.

What is a Table of Values?

A table of values contains two lists of numbers written alongside each other. The first list contains the chosen input values, which are often the 𝑥 coordinates. The second list contains the outputs obtained when this first list is put into a given equation, often the y coordinates. Together, the pairs of numbers in both lists make up coordinates that can be plotted as a graph.

In this example below, we have a table of values for y = 2𝑥 + 1.

There are two lists of numbers shown in the table of values.

The first list is the given 𝑥 values. Here we chose 0 to 4. We can choose any numbers we like to go in the 𝑥 row. We chose 0 to 4 because they are easy to substitute into equations because they are small numbers.

The y values have been calculated using the equation y = 2𝑥 + 1. This equation tells us to multiply the 𝑥 values by 2 and then add one to them to find the y values.

For example, when × = 3, we multiply 3 by 2 and then add 1.

3 × 2 + 1 = 7 and so, we write 7 in the y row below the 𝑥 value of 3.

Doing this calculation for each 𝑥 value results in a different number which is written in the y row alongside the 𝑥 value it came from.

How do you Make a Table of Values?

A table of values is made of two rows, the first labeled as 𝑥 and the second as y. In the 𝑥 row, consecutive numbers are chosen from between the smallest and largest 𝑥 coordinates on the axes given. The numbers in the y row are calculated by substituting these 𝑥 values into the given equation.

Here is a set of axes for plotting y = 2𝑥 – 3.

The smallest 𝑥 value is -10 and the largest 𝑥 value is 10.

We only need two points to draw a straight line but it is a good idea to choose three to five different 𝑥 values for the table of values. This allows us to better notice any mistakes and to help us draw a more accurate line through the points.

To choose what 𝑥 values to use for the table of values, it is best to choose small positive whole numbers if possible.

In this example, we will choose 0, 1, 2, 3 and 4.

The numbers in the y row are calculated by substituting these 𝑥 values into the equation y = 2𝑥 – 3. This tells us to multiply each 𝑥 value by 2 and then subtract 3.

The following table shows the calculations for the table of values.

How to Graph a Line from a Table of Values

A table of values contains pairs of 𝑥 and y values which form pairs of coordinates that can be plotted as points. The 𝑥 coordinate tells us how far right the point is and the y coordinate tells us how far up the point is. If the coordinates are negative, then the point is left or down respectively. Once each point is plotted, simply draw a line through them.

For example, here is the table of values for y = 3𝑥 – 5.

We have the following coordinates:

• (0, -5) from 𝑥 = 0 and y = -5
• (0, -2) from 𝑥 = 1 and y = -2
• (0, 1) from 𝑥 = 2 and y = 1
• (0, 4) from 𝑥 = 3 and y = 4
• (0, 7) from 𝑥 = 4 and y = 7

The coordinates are plotted on the graph as shown.

The following table explains how to plot coordinates:

• 𝑥 = 0 and y = -5 tells us to not go left or right and to go 5 down.
• 𝑥 = 1 and y = -2 tells us to go 1 right and 2 down.
• 𝑥 = 2 and y = 1 tells us to go right and 1 up.
• 𝑥 = 3 and y = 4 tells us to go 3 right and 4 up.
• 𝑥 = 4 and y = 7 tells us to go 4 right and 7 up

Each of these 2 numbers describe one point.

Once each point is plotted, the line is graphed by drawing a straight line through them all to the very edge of the axes.

How to Tell if a Table of Values is Linear

A table of values is linear if as 𝑥 increases by a constant amount, the y values all increase by a constant amount. If the y values increase by the same amount from one number to the next, then the coordinates will form a straight line when plotted. If the y values increase by different amounts each time, then the table of values is non-linear.

If something is linear, this means that it forms a straight line.

For example, here is a table of values where the y values increase by 2 every time that 𝑥 increases by 1.

Since the y values increase by 2 from one number to the next, this table of values is produced from a linear expression.

Alternatively, we can plot the 𝑥 and y values on a set of axes and look at it to see if it forms a straight line or not.

We can see that the coordinates form a straight line and so, the table of values produces a linear graph.

If the equation that produced the table of values contains only y and 𝑥 as the variables, then it will produce a linear graph. If it contains 𝑥² or any other powers of 𝑥, then it is non-linear.

This table of values is formed from the equation y= 2𝑥 + 1 and because there are no higher powers of 𝑥, the equation is a linear one.

How to Find the Equation of a Straight Line From a Table of Values

The equation of a straight line is y = m𝑥 + c, where m is the gradient and c is the y-intercept. The gradient is the amount y increases every time that 𝑥 increases by 1 in the table. The y-intercept is the y value that accompanies the 𝑥 = 0 value in the table.

Here is an example of finding the equation from a table of values.

The first step is to find the gradient. This is how much the y values increase every time that the 𝑥 values increase by 1.

We can see that the y values increase by 3 every time the 𝑥 values increase by 1. The gradient is 3.

The second step is to find the y-intercept. This is the y value that is below the 𝑥 = 0 value. Below 𝑥 = 0 is y = -5. Therefore the y-intercept is -5.

The equation of the line is given by y = m𝑥 + c, where m is the gradient and c is the y-intercept.

The gradient of 3 is written in front of the 𝑥 and the y-intercept of -5 is just written afterward.

The equation of the line is y = 3𝑥 – 5.