Proportional Geometry Examples in Math

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

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

Concept Recap

Proportional geometry studies how corresponding lengths, areas, and volumes scale between similar figures. If two triangles are similar with scale factor k, their sides are in ratio k, their areas in ratio k², and their volumes in ratio k³.

Similar triangles have proportional sides: if one side doubles, all sides double.

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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: When figures are similar, every length scales by $k$ , every area by $k^2$ , every volume by $k^3$ .

Common stuck point: The procedure for proportional geometry is the easy part; the trap is scaling area by $k$ instead of $k^2$ . Asking "Are the figures similar, and am I scaling a measurement by some power of the scale factor?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the figures similar, and am I scaling a measurement by some power of the scale factor?

Worked Examples

Example 1

easy

Two similar triangles have corresponding sides in the ratio 3:5. If the shorter triangle has a base of 9 cm, what is the base of the larger triangle?

Answer

15 cm

First step

Step 1: Set up the proportion:

\dfrac{\text{short base}}{\text{long base}} = \dfrac{3}{5}

Full solution

2
Step 2: $\dfrac{9}{x} = \dfrac{3}{5}$ .
3
Step 3: Cross-multiply: $3x = 45 \Rightarrow x = 15$ cm.

Similar figures have all corresponding lengths in the same ratio (the scale factor). Setting up a proportion and cross-multiplying is the standard method to find a missing length in similar figures.

Example 2

medium

A 6 ft tall person casts a shadow 4 ft long. At the same time, a tree casts a shadow 14 ft long. How tall is the tree?

Tree and its 14 ft shadow form a right triangle. Find the tree's height x.

Example 3

medium

Two similar rectangles have widths

4

cm and

10

cm. If the smaller rectangle has length

6

cm, find the length of the larger rectangle.

Larger similar rectangle (width = 10 cm). Find the unknown length x.

Example 4

medium

Triangle

ABC

is similar to triangle

DEF

with scale factor

k = 3

. If the area of

\triangle ABC

12

^2

, find the area of

\triangle DEF

Example 5

medium

Two similar cones have radii

2

cm and

5

cm. What is the ratio of their volumes?

Example 6

medium

Triangles

ABC

and

DEF

are similar with

AB/DE = 4/9

. If

BC = 12

, find

EF

Triangle DEF with DE = 9. Given AB/DE = 4/9 and BC = 12, find EF = x.

Example 7

medium

A map's scale is

1 : 25000

. Two landmarks are

8

cm apart on the map. What is the actual distance in km?

Example 8

hard

Two similar cylinders have lateral surface areas in the ratio

9 : 49

. If the smaller has volume

54

^3

, find the volume of the larger.

Example 9

hard

In triangle

ABC

, point

D

AB

and point

E

AC

make

DE \parallel BC

with

AD = 4

and

DB = 6

. If

AE = 3

, find

EC

Example 10

hard

Two similar triangles have hypotenuses

13

and

39

. The smaller triangle has a leg of length

5

. Find the corresponding leg of the larger.

Larger similar right triangle with hypotenuse = 39. The smaller has hypotenuse 13 and leg 5. Find leg x.

Example 11

hard

Two similar pyramids have heights

5

cm and

12

cm. If the smaller has surface area

50\pi

^2

, find the surface area of the larger.

Example 12

challenge

Two similar triangles have areas

A_1

and

A_2

with

A_2 = 4 A_1

. A median of the first triangle has length

m_1 = 6

. Find the corresponding median of the second.

Example 13

challenge

A model boat's sail catches wind force proportional to sail area. The real boat is

10

times as long. By what factor does the sail force scale? If sail force must equal weight (volume) for the boat to capsize, why do big boats handle wind better proportionally?

Practice Problems

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

Example 1

easy

Two similar rectangles have widths 5 cm and 15 cm. What is the scale factor from the smaller to the larger?

Example 2

hard

On a map, 1 cm represents 50 km. Two cities are 3.6 cm apart on the map. What is the actual distance? Also, a lake has an area of 4 cm² on the map. What is its actual area in km²?

Example 3

easy

In similar triangles, corresponding sides are in ratio

3:6

. Simplify this ratio.

Example 4

easy

Similar triangles have scale factor

2

. A side of

5

in the small one corresponds to what in the large?

Example 5

easy

Solve the proportion

\frac{3}{4} = \frac{x}{8}

Example 6

easy

In similar figures, are corresponding angles equal or proportional?

Example 7

easy

6

-ft person casts a

9

-ft shadow. Set up the height-to-shadow ratio.

Person 6 ft tall casts a 9 ft shadow. The height-to-shadow ratio is 6:9 = 2:3.

Example 8

easy

Similar triangles have side ratio

1:3

. What is the ratio of their areas?

Example 9

easy

A map scale is

1:1000

. A wall is

5

cm on the map. How long is it in real life (cm)?

Example 10

easy

Solve

\frac{x}{12} = \frac{2}{3}

Example 11

medium

Triangle

ABC \sim

triangle

DEF

AB = 6

BC = 8

DE = 9

. Find

EF

Triangle DEF similar to ABC (AB=6, BC=8, DE=9). Find EF = x.

Example 12

medium

5

-ft pole casts a

2

-ft shadow. A nearby flagpole casts a

14

-ft shadow. Find the flagpole's height.

Flagpole and its 14 ft shadow. A 5 ft pole casts a 2 ft shadow. Find the flagpole height x.

Example 13

medium

Two similar pentagons have perimeters

20

and

50

. Find the ratio of their areas.

Example 14

medium

In triangle

ABC

DE \parallel BC

with

D

AB

E

AC

. If

AD = 3

AB = 9

, and

AE = 5

, find

AC

Example 15

medium

Two similar solids have volumes

8

and

27

. Find the ratio of their surface areas.

Example 16

medium

Why must you match sides by their position (relative to equal angles) when writing a proportion for similar triangles?

Example 17

medium

A model bridge is

\tfrac{1}{50}

scale. The real bridge's deck has area

200\,\text{m}^2

. Find the model deck's area.

Example 18

medium

Two similar triangles have areas

9

and

16

. The smaller has a side of length

6

. Find the corresponding side of the larger.

Example 19

challenge

In a right triangle, the altitude to the hypotenuse has length

h

, splitting the hypotenuse into segments

p

and

q

. Show

h^2 = pq

using similar triangles.

Example 20

challenge

A cone is cut by a plane parallel to its base, halfway up. Find the ratio of the small top cone's volume to the whole cone's volume.

Example 21

challenge

Two similar triangles have areas in ratio

2:3

. The perimeter of the larger is

30

. Find the perimeter of the smaller.

Example 22

challenge

Explain why all the dimensional ratios (length, area, volume) between two similar figures are powers of a single number

k

Example 23

easy

Two similar triangles have a scale factor of

4

. If the smaller triangle has a side of length

7

, what is the corresponding side of the larger?

Example 24

easy

Solve the proportion

\frac{x}{12} = \frac{5}{8}

for

x

Example 25

easy

Two similar triangles have sides in ratio

2:7

. What is the ratio of their perimeters?

Example 26

easy

Two similar squares have sides

4

and

10

. What is the ratio of their areas?

Example 27

medium

A photograph

4

in by

6

in is enlarged so the longer side becomes

15

in. What is the new short side?

Example 28

medium

On a blueprint,

\frac{1}{4}

inch represents

1

foot. A room is

18

feet long. How long is the room on the blueprint?

Example 29

medium

A small triangle has area

9

^2

and is similar to a larger triangle with area

144

^2

. What is the scale factor of their sides?

Example 30

medium

Two similar prisms have volumes

27

^3

and

216

^3

. Find the ratio of their surface areas.

Example 31

medium

Two similar pentagons have perimeters

30

and

48

. The smaller has area

50

. Find the area of the larger.

Example 32

hard

A scale model of a car has volume

1

liter. The real car has volume

216

^3

. Note

1

^3 = 1000

liters. What is the linear scale factor of the model?

Example 33

hard

A triangular sail is similar to another sail with

\frac{1}{3}

the height. If the bigger sail uses

9

^2

of fabric, how much does the smaller sail use?

Example 34

hard

If a recipe is scaled up so all ingredient quantities triple, by what factor does the cooking pot volume scale (assuming geometric similarity)? Then by what factor must the linear size of the pot grow?

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Related Concepts

Background Knowledge

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

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