Similar shapes and scale factors

Lesson overview

Geometry — similarity and enlargement

Two shapes are **similar** when they have the same shape but different sizes. All corresponding angles are equal, and all corresponding sides are in the same ratio. The **scale factor of enlargement** $k$ tells you how much bigger one shape is than the other. If two shapes are similar: - Corresponding lengths are multiplied by $k$ - Corresponding areas are multiplied by $k^2$ - Corresponding volumes are multiplied by $k^3$ $\text{Scale factor} = \frac{\text{new length}}{\text{original length}}$

Similar triangles have the same angles and proportional sides

Triangle A has sides $3\text{ cm}$ , $4\text{ cm}$ , $5\text{ cm}$ . Triangle B is similar with scale factor $2$ . Find the sides of Triangle B.

Triangle A with sides 3, 4, 5 cm

Scale factor $k = 2$

New base $3 \times 2 = 6$ cm

New height $4 \times 2 = 8$ cm

Multiply every length by the scale factor $k$ .

Area scales by $k^2$ , volume by $k^3$

If a shape is enlarged with scale factor $k = 3$ , then the area is multiplied by $k^2 = 9$ and the volume by $k^3 = 27$ .

Lengths $\times k$

scale factor

Area $\times k^2$

squared

Volume $\times k^3$

cubed

Remember: area is two-dimensional, volume is three-dimensional.

Finding the scale factor from two similar shapes

Two similar rectangles have corresponding sides of $6\text{ cm}$ and $9\text{ cm}$ . The scale factor is:

Scale factor $=$ corresponding new length $\div$ corresponding original length.

Worked example

Two similar cylinders have radii $4\text{ cm}$ and $6\text{ cm}$ . The smaller cylinder has volume $200\text{ cm}^3$ . Find the volume of the larger cylinder.

Loading 3D viewer…

Find the scale factor: $k = \dfrac{6}{4} = 1.5$ .
The volume scale factor is $k^3 = 1.5^3 = 3.375$ .
Volume of larger cylinder $= 200 \times 3.375 = 675\text{ cm}^3$ .
Check: the larger cylinder is $1.5$ times bigger in every linear dimension, so the volume should be much more than double.

Try it

Identify the scale factor first, then decide whether you need $k$, $k^2$, or $k^3$.

Question 1

Two similar triangles have corresponding sides of $5\text{ cm}$ and $12\text{ cm}$ . What is the scale factor of enlargement?

Question 2

A shape is enlarged with scale factor $3$ . If the original area is $10\text{ cm}^2$ , what is the new area?

Question 3

Two similar cubes have volumes $64\text{ cm}^3$ and $512\text{ cm}^3$ . What is the scale factor of the lengths?

Question 4

Which statement about similar shapes is correct?

Question 5

A rectangle has area $15\text{ cm}^2$ . It is enlarged with scale factor $4$ . What is the area of the enlarged rectangle?

Common mistakes

Watch for these when working through the lesson.

Using the area scale factor $k^2$ when the question asks about lengths.
Using the length scale factor $k$ when the question asks about areas or volumes.
Confusing similar with congruent — similar shapes are the same shape but different sizes, congruent shapes are identical in size and shape.

Similar shapes and scale factors

Lesson overview

Worked example

Try it

Common mistakes

Related topics

Independent practice will plug in here