Practice Proportional Geometry in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy
In similar triangles, corresponding sides are in ratio 3:63:6. Simplify this ratio.

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

hard
Two similar triangles have hypotenuses 1313 and 3939. The smaller triangle has a leg of length 55. Find the corresponding leg of the larger.

Example 4

medium
Two similar solids have volumes 88 and 2727. Find the ratio of their surface areas.

Example 5

medium
A map's scale is 1:250001 : 25000. Two landmarks are 88 cm apart on the map. What is the actual distance in km?

Example 6

medium
Triangle ABCABC is similar to triangle DEFDEF with scale factor k=3k = 3. If the area of △ABC\triangle ABC is 1212 cm2^2, find the area of △DEF\triangle DEF.

Example 7

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?

Example 8

medium
A small triangle has area 99 cm2^2 and is similar to a larger triangle with area 144144 cm2^2. What is the scale factor of their sides?

Example 9

easy
Two similar triangles have sides in ratio 2:72:7. What is the ratio of their perimeters?

Example 10

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 11

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?

Example 12

easy
In similar figures, are corresponding angles equal or proportional?

Example 13

medium
A flagpole's shadow is 2424 ft and a nearby 44-ft post casts a 33-ft shadow at the same time. Find the flagpole height.

Example 14

medium
A 55-ft pole casts a 22-ft shadow. A nearby flagpole casts a 1414-ft shadow. Find the flagpole's height.

Example 15

medium
On a blueprint, 14\frac{1}{4} inch represents 11 foot. A room is 1818 feet long. How long is the room on the blueprint?

Example 16

challenge
Two similar triangles have areas in ratio 2:32:3. The perimeter of the larger is 3030. Find the perimeter of the smaller.

Example 17

medium
Two similar cones have radii 22 cm and 55 cm. What is the ratio of their volumes?

Example 18

hard
Two similar cylinders have lateral surface areas in the ratio 9:499 : 49. If the smaller has volume 5454 cm3^3, find the volume of the larger.

Example 19

easy
Solve x12=23\frac{x}{12} = \frac{2}{3}.

Example 20

medium
A model bridge is 150\tfrac{1}{50} scale. The real bridge's deck has area 200 m2200\,\text{m}^2. Find the model deck's area.
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