Practice Scaling in Space in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

How length, area, and volume measurements change when a figure is uniformly enlarged or shrunk by a scale factor.

Double the size: length ร—2\times 2, area ร—4\times 4, volume ร—8\times 8.

Showing a random 20 of 50 problems.

Example 1

medium
Two similar cans have volumes 5454 and 128128. Find the ratio of their heights.

Example 2

medium
A recipe is doubled in every linear dimension of a cake pan. How much more batter is needed?

Example 3

challenge
Two similar solid balls of the same material have masses 2727 kg and 125125 kg. Find the ratio of their surface areas.

Example 4

easy
A cube has volume 1ย cm31\text{ cm}^3. Scale lengths by 44. Find the new volume.

Example 5

easy
Volume scales by what power of the linear scale factor kk?

Example 6

medium
Two similar boxes have surface areas 5050 and 200200. The larger holds 640640 cm3^3. How much does the smaller hold?

Example 7

challenge
Explain why doubling a pizza's diameter more than doubles the food you get, and quantify it.

Example 8

hard
Why does a small ice cube melt faster than a large block of the same shape?

Example 9

medium
A sphere's surface area grows from 100100 to 400400 cm2^2. By what factor did its radius scale, and what is the volume factor?

Example 10

challenge
Why can an ant carry many times its body weight, but a scaled-up 'giant ant' could not support itself? Use scaling.

Example 11

medium
Two similar cones have heights 44 and 77. Find the ratio of their volumes.

Example 12

medium
A sphere's radius is tripled. By what factor does its surface area increase, and by what factor its volume?

Example 13

hard
Two similar pyramids have heights 4 m and 10 m. If the smaller pyramid has volume 3232 mยณ, what is the volume of the larger?

Example 14

easy
A rectangle has area 2424. All lengths are scaled by 55. Find the new area.

Example 15

medium
A sphere has radius 2 cm and volume V=43ฯ€r3V = \frac{4}{3}\pi r^3. If the radius is tripled, how many times larger is the new volume?

Example 16

easy
Scale a solid's lengths by 22. By what factor does its volume change?

Example 17

medium
A statue is scaled up by factor 44. Its original weight (proportional to volume) was 55 kg. Find the new weight.

Example 18

hard
Strength of a bone scales with cross-sectional area (k2k^2). Body weight scales with volume (k3k^3). Show that for a uniformly scaled animal, relative strength (strength/weight) declines as 1/k1/k.

Example 19

easy
If a figure's area increases by 3636 times, by what factor did the lengths scale?

Example 20

easy
A cube of volume 22 has its sides doubled. Find the new volume.
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