Similar Figures Examples: 22 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Similar Figures.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Similar figures have the same shape with corresponding angles equal and corresponding sides proportional.

One figure is an enlarged or reduced copy of another—same shape, same angles, but possibly different size.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Similar figures are scaled copies: corresponding angles match exactly and corresponding sides share one constant ratio.

Common stuck point: The procedure for similar figures is the easy part; the trap is treating similar as congruent. Asking "Do the two figures have all corresponding angles equal and all corresponding sides in one common ratio?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do the two figures have all corresponding angles equal and all corresponding sides in one common ratio?

Worked Examples

Example 1

easy

Two similar triangles have corresponding sides

6

cm and

10

cm. If the smaller triangle has a perimeter of

21

cm, find the perimeter of the larger triangle.

Smaller similar triangle (one corresponding side = 6 cm)

Answer

The larger triangle has perimeter

35

cm.

First step

Scale factor (larger to smaller):

k = \dfrac{10}{6} = \dfrac{5}{3}

Full solution

2
Perimeters of similar figures scale by the same ratio as corresponding sides.
3
Perimeter of larger triangle $= 21 \times \dfrac{5}{3} = 35$ cm.

All linear measurements in similar figures scale by the same ratio

k

. Only areas scale by

k^2

and volumes by

k^3

. Once the scale factor is identified from one pair of corresponding sides, all other proportions follow.

Example 2

hard

Rectangle

ABCD \sim

Rectangle

EFGH

with

AB = 8

BC = 5

EF = 12

. Find

FG

and the ratio of areas.

Rectangle ABCD (AB = 8, BC = 5); find FG in the similar rectangle EFGH with EF = 12

Example 3

medium

Two similar polygons have perimeters

36

and

60

. The smaller has a side of length

9

. Find the matching side on the larger.

Example 4

medium

Two similar triangles have areas

32

and

50

. The smaller has a side

12

. Find the matching side on the larger.

Example 5

hard

A small statue is similar to a big statue. The big one is

3

times as tall and weighs

135

kg. Estimate the small statue's weight assuming same density.

Example 6

challenge

Two similar prisms have volumes in ratio

27:64

and total surface areas

a

and

b

. If

a=108

^2

, find

b

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Triangle

PQR \sim

Triangle

STU

PQ = 4

ST = 6

QR = 7

. Find

TU

Triangle PQR (PQ = 4, QR = 7); find TU in similar triangle STU where ST

Example 2

medium

Two similar cones have base radii

3

cm and

7

cm. What is the ratio of their volumes?

Example 3

easy

Two similar triangles have corresponding sides

4

and

10

. Find the scale factor (large to small).

Example 4

easy

Two similar pentagons have scale factor

4

. If a side on the small one is

3

, find the matching side on the large one.

Example 5

easy

Two similar triangles have side ratio

3:5

. Find the perimeter ratio.

Example 6

easy

Two similar squares have side ratio

1:6

. Find the area ratio.

Example 7

medium

Triangle

ABC\sim

Triangle

DEF

with

AB=5

DE=20

. If

AC=8

, find

DF

Example 8

medium

Two similar rectangles have areas

48

and

108

. Find the linear scale factor (large to small).

Example 9

medium

Two similar pyramids have heights

6

cm and

9

cm. Find the ratio of their surface areas.

Example 10

medium

Two similar cylinders have radii

3

and

5

. Find the ratio of their lateral surface areas.

Example 11

medium

A photograph is enlarged so that the new perimeter is

3

times the old. By what factor does the area grow?

Example 12

hard

Two similar solids have surface areas

48

and

108

. Find the ratio of their volumes.

Example 13

hard

A scale model of a building uses scale

1:50

. The real building has

2{,}500

^2

of glass. How much glass does the model need?

Example 14

hard

Two similar cones have volumes

54\pi

and

128\pi

. Find the ratio of their heights.

Example 15

hard

Triangle

ABC

has

AB=6, BC=8, CA=10

. A similar triangle has its longest side equal to

25

. Find the perimeter of the similar triangle.

Example 16

challenge

A model airplane (scale

1:48

) has wings totaling

0.5

^2

in area. Find the real airplane's wing area in m

^2

← Back to Similar Figures Practice Mode

Related Concepts

Similarity Proportions Scale Drawings

Background Knowledge

These ideas may be useful before you work through the harder examples.

similarityproportionsscale drawings