Similar Shapes (Area and Volume)

What you'll learn

How to recognise and use mathematically similar solid shapes.
How length, surface area and volume change using scale factors.
How to work backwards from surface area to length, or from area to volume.
How to solve frustum questions by subtracting cone volumes.

1. The key idea: similar solids

Two 3D shapes are mathematically similar if they have exactly the same shape, but may be different sizes. All matching lengths are enlarged or reduced by the same multiplier.

Definition

Scale factor

The linear scale factor is the multiplier from one length on a shape to the matching length on a similar shape.

For example, if a radius changes from 4 cm to 10 cm, the linear scale factor is:

104=2.5\frac{10}{4} = 2.5

The matching parts are called corresponding lengths. For example, radius matches radius, height matches height, diameter matches diameter, and length matches length.

Here is the big picture: lengths scale once, areas scale twice, and volumes scale three times.

Diagram showing similar shapes with length scale factor k, area scale factor k squared, and volume scale factor k cubed

Key Idea

The three scale factors

If the linear scale factor is $k$ , then:

Area scale factor is $k^2$
Volume scale factor is $k^3$

Example

Finding the surface area of a larger similar cone

Two similar cones have base radii 5 cm and 15 cm. The smaller cone has total surface area 70 cm². Find the total surface area of the larger cone.

Find the linear scale factor from the smaller cone to the larger cone:
$\frac{15}{5} = 3$
Square the linear scale factor to get the area scale factor:
$k^2 = 3^2 = 9$
Multiply the smaller surface area by the area scale factor:
$70 \times 9 = 630$
The surface area of the larger cone is 630 cm².

Common Mistake

Using the length scale factor for area

If a length doubles, the surface area does not double. It becomes 4 times as large, because $2^2 = 4$ .

2. Surface area uses the square of the scale factor

Surface area is a 2D measurement wrapped around a 3D object. That means it depends on two length directions at once.

For similar shapes:

factor=k2\text{area scale factor} = k^2

This works for curved surface area, total surface area, or the area of a matching face, as long as the shapes are similar.

Example

A cylinder with double the height

Two similar cylinders are shown. Cylinder B is twice as tall as cylinder A. Cylinder A has total surface area 180 cm². Find the total surface area of cylinder B.

Because the cylinders are similar, if the height doubles, every length doubles. So the linear scale factor is:
$k = 2$
Find the area scale factor:
$k^2 = 2^2 = 4$
Multiply the surface area of cylinder A by 4:
$180 \times 4 = 720$
The total surface area of cylinder B is 720 cm².

Tip

Check the direction

Before multiplying, ask: “Am I going from smaller to larger, or larger to smaller?” If you go from larger to smaller, you divide by the scale factor instead.

3. Volume uses the cube of the scale factor

Volume is a 3D measurement, so it depends on three length directions. That is why volume changes by the cube of the linear scale factor.

For similar shapes:

factor=k3\text{volume scale factor} = k^3

Example

Finding the volume of a larger similar cylinder

Two similar cylinders have lengths 4 cm and 6 cm. The smaller cylinder has volume 80 cm³. Find the volume of the larger cylinder.

Find the linear scale factor from the smaller cylinder to the larger cylinder:
$\frac{6}{4} = \frac{3}{2}$
Cube the linear scale factor to get the volume scale factor:
$k3=(32)3=278k^3 = \left(\frac{3}{2}\right)^3 = \frac{27}{8}$
Multiply the smaller volume by the volume scale factor:
$80 \times \frac{27}{8} = 270$
The volume of the larger cylinder is 270 cm³.

Common Mistake

Squaring instead of cubing for volume

For volume, do not use $k^2$ . Surface area uses $k^2$ , but volume uses $k^3$ .

4. Working backwards from area to length

Sometimes you are given two surface areas and need to find a missing length. In that case, start by finding the area scale factor, then square root it to get the linear scale factor.

Key Idea

Area back to length

If the area scale factor is known, then:

\sqrt{\text{area scale factor}}

Example

Using surface areas to find a missing height

Two similar cones P and Q have total surface areas 30 cm² and 120 cm². Cone P has height 5 cm. Find the height of cone Q.

Find the area scale factor from P to Q:
$12030=4\frac{120}{30} = 4$
Square root the area scale factor to find the linear scale factor:
$\sqrt{4} = 2$
Multiply the height of cone P by the linear scale factor:
$\times 2 = 10$
The height of cone Q is 10 cm.

Tip

Square root for lengths

When you are given areas but asked for a length, remember: area to length means square root.

5. Linking surface area and volume

A common exam style is: you are given the surface areas of two similar solids, plus the volume of one, and you need the volume of the other.

The method is:

Use the surface areas to find the area scale factor.
Square root to get the linear scale factor.
Cube to get the volume scale factor.
Multiply or divide the volume.

Example

Finding a volume from two surface areas

Two geometrically similar solids X and Y have surface areas 500 cm² and 1125 cm². The volume of X is 1600 cm³. Find the volume of Y.

Find the area scale factor from X to Y:
$1125500=2.25\frac{1125}{500} = 2.25$
Square root to find the linear scale factor:
$\sqrt{2.25} = 1.5$
Cube the linear scale factor to find the volume scale factor:
$k^3 = 1.5^3 = 3.375$
Multiply the volume of X by the volume scale factor:
$1600 \times 3.375 = 5400$
The volume of Y is 5400 cm³.

Common Mistake

Cubing the area scale factor

Do not cube 2.25 in this example. The 2.25 is already an area scale factor, not a length scale factor. Square root first, then cube.

6. Frustums: subtracting similar cone volumes

A frustum is what is left when the top of a cone is cut off parallel to the base. In many IGCSE questions, you find the frustum volume by doing:

volume\text{frustum volume} = \text{large cone volume} - \text{small cone volume}

Definition

Frustum

A frustum is the solid made when a cone or pyramid has its top cut off parallel to its base.

This diagram shows the subtraction idea: the frustum is the large cone with the small cone removed.

Diagram of a large cone split into a small cone and a frustum, with dimensions and subtraction formula

To calculate cone volume, use:

\frac{1}{3}\pi r^2h

where $r$ is the radius and $h$ is the perpendicular height.

Example

Volume of a frustum in terms of pi

A large cone has height 9 cm and base diameter 12 cm. A smaller cone at the top has height 3 cm and base diameter 4 cm. Find the volume of the frustum left behind, in terms of pi.

Work out the radius of the large cone. The diameter is 12 cm, so the radius is 6 cm.
Find the volume of the large cone:
$\frac{1}{3}\pi \times 6^2 \times 9 = 108\pi$
Work out the radius of the small cone. The diameter is 4 cm, so the radius is 2 cm.
Find the volume of the small cone:
$\frac{1}{3}\pi \times 2^2 \times 3 = 4\pi$
Subtract the small cone from the large cone:
$108π−4π=104π108\pi - 4\pi = 104\pi$
The volume of the frustum is 104π cm³.

Common Mistake

Use radius, not diameter

The cone volume formula uses radius. If you are given a diameter, halve it before substituting into $\frac{1}{3}\pi r^2h$ .

7. Choosing the right multiplier

Here is a quick decision guide:

Asked for a length? Use $k$ .
Asked for an area or surface area? Use $k^2$ .
Asked for a volume? Use $k^3$ .
Given areas but need length? Use a square root.
Given volumes but need length? Use a cube root.

Example

Two-part similar cylinder question

Two similar cylinders P and Q have total surface areas 100π cm² and 900π cm². Cylinder P has length 6 cm and volume 150π cm³. Find the length and volume of cylinder Q.

Find the area scale factor from P to Q:
$900π100π=9\frac{900\pi}{100\pi} = 9$
Square root to get the linear scale factor:
$\sqrt{9} = 3$
Use the linear scale factor to find the length of Q:
$\times 3 = 18$
Cube the linear scale factor to find the volume scale factor:
$k^3 = 3^3 = 27$
Multiply the volume of P by 27:
$150π×27=4050π150\pi \times 27 = 4050\pi$
The length of Q is 18 cm and the volume of Q is 4050π cm³.

Exam technique

In the exam

Write down the linear scale factor first whenever possible, and label its direction clearly.
If you are given surface areas, square root before using it on lengths or cubing it for volumes.
Check whether the question wants an exact answer in terms of π, and do not round if it does.

Self review

Check yourself

If the length scale factor is 4, what are the area and volume scale factors?
If two similar solids have surface areas in the ratio 25:64, what is the length ratio?
In a frustum question, which two cone volumes do you subtract?

Similar Shapes (Area and Volume)

What you'll learn

How to recognise and use mathematically similar solid shapes.
How length, surface area and volume change using scale factors.
How to work backwards from surface area to length, or from area to volume.
How to solve frustum questions by subtracting cone volumes.

1. The key idea: similar solids

Two 3D shapes are mathematically similar if they have exactly the same shape, but may be different sizes. All matching lengths are enlarged or reduced by the same multiplier.

Definition

Scale factor

The linear scale factor is the multiplier from one length on a shape to the matching length on a similar shape.

For example, if a radius changes from 4 cm to 10 cm, the linear scale factor is:

104=2.5\frac{10}{4} = 2.5

The matching parts are called corresponding lengths. For example, radius matches radius, height matches height, diameter matches diameter, and length matches length.

Here is the big picture: lengths scale once, areas scale twice, and volumes scale three times.

Diagram showing similar shapes with length scale factor k, area scale factor k squared, and volume scale factor k cubed

Key Idea

The three scale factors

If the linear scale factor is $k$ , then:

Area scale factor is $k^2$
Volume scale factor is $k^3$

Example

Finding the surface area of a larger similar cone

Two similar cones have base radii 5 cm and 15 cm. The smaller cone has total surface area 70 cm². Find the total surface area of the larger cone.

Find the linear scale factor from the smaller cone to the larger cone:
$\frac{15}{5} = 3$
Square the linear scale factor to get the area scale factor:
$k^2 = 3^2 = 9$
Multiply the smaller surface area by the area scale factor:
$70 \times 9 = 630$
The surface area of the larger cone is 630 cm².

Common Mistake

Using the length scale factor for area

If a length doubles, the surface area does not double. It becomes 4 times as large, because $2^2 = 4$ .

2. Surface area uses the square of the scale factor

Surface area is a 2D measurement wrapped around a 3D object. That means it depends on two length directions at once.

For similar shapes:

factor=k2\text{area scale factor} = k^2

This works for curved surface area, total surface area, or the area of a matching face, as long as the shapes are similar.

Example

A cylinder with double the height

Two similar cylinders are shown. Cylinder B is twice as tall as cylinder A. Cylinder A has total surface area 180 cm². Find the total surface area of cylinder B.

Because the cylinders are similar, if the height doubles, every length doubles. So the linear scale factor is:
$k = 2$
Find the area scale factor:
$k^2 = 2^2 = 4$
Multiply the surface area of cylinder A by 4:
$180 \times 4 = 720$
The total surface area of cylinder B is 720 cm².

Tip

Check the direction

Before multiplying, ask: “Am I going from smaller to larger, or larger to smaller?” If you go from larger to smaller, you divide by the scale factor instead.

3. Volume uses the cube of the scale factor

Volume is a 3D measurement, so it depends on three length directions. That is why volume changes by the cube of the linear scale factor.

For similar shapes:

factor=k3\text{volume scale factor} = k^3

Example

Finding the volume of a larger similar cylinder

Two similar cylinders have lengths 4 cm and 6 cm. The smaller cylinder has volume 80 cm³. Find the volume of the larger cylinder.

Find the linear scale factor from the smaller cylinder to the larger cylinder:
$\frac{6}{4} = \frac{3}{2}$
Cube the linear scale factor to get the volume scale factor:
$k3=(32)3=278k^3 = \left(\frac{3}{2}\right)^3 = \frac{27}{8}$
Multiply the smaller volume by the volume scale factor:
$80 \times \frac{27}{8} = 270$
The volume of the larger cylinder is 270 cm³.

Common Mistake

Squaring instead of cubing for volume

For volume, do not use $k^2$ . Surface area uses $k^2$ , but volume uses $k^3$ .

4. Working backwards from area to length

Sometimes you are given two surface areas and need to find a missing length. In that case, start by finding the area scale factor, then square root it to get the linear scale factor.

Key Idea

Area back to length

If the area scale factor is known, then:

\sqrt{\text{area scale factor}}

Example

Using surface areas to find a missing height

Two similar cones P and Q have total surface areas 30 cm² and 120 cm². Cone P has height 5 cm. Find the height of cone Q.

Find the area scale factor from P to Q:
$12030=4\frac{120}{30} = 4$
Square root the area scale factor to find the linear scale factor:
$\sqrt{4} = 2$
Multiply the height of cone P by the linear scale factor:
$\times 2 = 10$
The height of cone Q is 10 cm.

Tip

Square root for lengths

When you are given areas but asked for a length, remember: area to length means square root.

5. Linking surface area and volume

A common exam style is: you are given the surface areas of two similar solids, plus the volume of one, and you need the volume of the other.

The method is:

Use the surface areas to find the area scale factor.
Square root to get the linear scale factor.
Cube to get the volume scale factor.
Multiply or divide the volume.

Example

Finding a volume from two surface areas

Two geometrically similar solids X and Y have surface areas 500 cm² and 1125 cm². The volume of X is 1600 cm³. Find the volume of Y.

Find the area scale factor from X to Y:
$1125500=2.25\frac{1125}{500} = 2.25$
Square root to find the linear scale factor:
$\sqrt{2.25} = 1.5$
Cube the linear scale factor to find the volume scale factor:
$k^3 = 1.5^3 = 3.375$
Multiply the volume of X by the volume scale factor:
$1600 \times 3.375 = 5400$
The volume of Y is 5400 cm³.

Common Mistake

Cubing the area scale factor

Do not cube 2.25 in this example. The 2.25 is already an area scale factor, not a length scale factor. Square root first, then cube.

6. Frustums: subtracting similar cone volumes

A frustum is what is left when the top of a cone is cut off parallel to the base. In many IGCSE questions, you find the frustum volume by doing:

volume\text{frustum volume} = \text{large cone volume} - \text{small cone volume}

Definition

Frustum

A frustum is the solid made when a cone or pyramid has its top cut off parallel to its base.

This diagram shows the subtraction idea: the frustum is the large cone with the small cone removed.

Diagram of a large cone split into a small cone and a frustum, with dimensions and subtraction formula

To calculate cone volume, use:

\frac{1}{3}\pi r^2h

where $r$ is the radius and $h$ is the perpendicular height.

Example

Volume of a frustum in terms of pi

A large cone has height 9 cm and base diameter 12 cm. A smaller cone at the top has height 3 cm and base diameter 4 cm. Find the volume of the frustum left behind, in terms of pi.

Work out the radius of the large cone. The diameter is 12 cm, so the radius is 6 cm.
Find the volume of the large cone:
$\frac{1}{3}\pi \times 6^2 \times 9 = 108\pi$
Work out the radius of the small cone. The diameter is 4 cm, so the radius is 2 cm.
Find the volume of the small cone:
$\frac{1}{3}\pi \times 2^2 \times 3 = 4\pi$
Subtract the small cone from the large cone:
$108π−4π=104π108\pi - 4\pi = 104\pi$
The volume of the frustum is 104π cm³.

Common Mistake

Use radius, not diameter

The cone volume formula uses radius. If you are given a diameter, halve it before substituting into $\frac{1}{3}\pi r^2h$ .

7. Choosing the right multiplier

Here is a quick decision guide:

Asked for a length? Use $k$ .
Asked for an area or surface area? Use $k^2$ .
Asked for a volume? Use $k^3$ .
Given areas but need length? Use a square root.
Given volumes but need length? Use a cube root.

Example

Two-part similar cylinder question

Two similar cylinders P and Q have total surface areas 100π cm² and 900π cm². Cylinder P has length 6 cm and volume 150π cm³. Find the length and volume of cylinder Q.

Find the area scale factor from P to Q:
$900π100π=9\frac{900\pi}{100\pi} = 9$
Square root to get the linear scale factor:
$\sqrt{9} = 3$
Use the linear scale factor to find the length of Q:
$\times 3 = 18$
Cube the linear scale factor to find the volume scale factor:
$k^3 = 3^3 = 27$
Multiply the volume of P by 27:
$150π×27=4050π150\pi \times 27 = 4050\pi$
The length of Q is 18 cm and the volume of Q is 4050π cm³.

Exam technique

In the exam

Write down the linear scale factor first whenever possible, and label its direction clearly.
If you are given surface areas, square root before using it on lengths or cubing it for volumes.
Check whether the question wants an exact answer in terms of π, and do not round if it does.

Self review

Check yourself

If the length scale factor is 4, what are the area and volume scale factors?
If two similar solids have surface areas in the ratio 25:64, what is the length ratio?
In a frustum question, which two cone volumes do you subtract?

Similar Shapes (Area and Volume)

What you'll learn

1. The key idea: similar solids

2. Surface area uses the square of the scale factor

3. Volume uses the cube of the scale factor

4. Working backwards from area to length

5. Linking surface area and volume

6. Frustums: subtracting similar cone volumes

7. Choosing the right multiplier

Recap questions

You've reached the end

Similar Shapes (Area and Volume)

What you'll learn

1. The key idea: similar solids

2. Surface area uses the square of the scale factor

3. Volume uses the cube of the scale factor

4. Working backwards from area to length

5. Linking surface area and volume

6. Frustums: subtracting similar cone volumes

7. Choosing the right multiplier

Recap questions

You've reached the end

Similar Shapes (Area and Volume)

What you'll learn

1. The key idea: similar solids

2. Surface area uses the square of the scale factor

3. Volume uses the cube of the scale factor

4. Working backwards from area to length

5. Linking surface area and volume

6. Frustums: subtracting similar cone volumes

7. Choosing the right multiplier

Recap questions

You've reached the end

Similar Shapes (Area and Volume) Revision Guide

Similar Shapes (Area and Volume)

What you'll learn

1. The key idea: similar solids

2. Surface area uses the square of the scale factor

3. Volume uses the cube of the scale factor

4. Working backwards from area to length

5. Linking surface area and volume

6. Frustums: subtracting similar cone volumes

7. Choosing the right multiplier

Recap questions

You've reached the end

Similar Shapes (Area and Volume) Revision Guide