Scaling in Space Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Scaling in Space.

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

Concept Recap

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

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

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: When a figure is enlarged by a scale factor $k$ , lengths grow by $k$ , areas by $k^2$ , and volumes by $k^3$ .

Common stuck point: The procedure for scaling in space is the easy part; the trap is scaling area by $k$ instead of $k^2$ . Asking "Is a whole figure resized by one factor, and do I need to scale a measure by the right power of it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is a whole figure resized by one factor, and do I need to scale a measure by the right power of it?

Worked Examples

Example 1

easy

A square has side length 3 cm. If all lengths are doubled (scale factor

k=2

), what are the new perimeter and area?

Answer

New perimeter = 24 cm; new area = 36 cm².

First step

Step 1: Original side = 3 cm. New side

= 3 \times 2 = 6

cm.

Full solution

2
Step 2: Perimeter scales by $k$ : new perimeter $= 4 \times 6 = 24$ cm (original was $12$ cm, doubled).
3
Step 3: Area scales by $k^2$ : original area $= 9$ cm², new area $= 9 \times 4 = 36$ cm².
4
Step 4: Verify: $6^2 = 36$ cm².

When a shape is scaled by factor

k

: all lengths multiply by

k

, all areas multiply by

k^2

, and all volumes multiply by

k^3

. This is because area is two-dimensional (two lengths multiplied) and volume is three-dimensional.

Example 2

medium

A sphere has radius 2 cm and volume

V = \frac{4}{3}\pi r^3

. If the radius is tripled, how many times larger is the new volume?

Example 3

medium

A statue is

\tfrac{1}{4}

scale of the original (scale factor

k = \tfrac{1}{4}

). The original needs

80

kg of bronze. How much does the scale model need (same density)?

Practice Problems

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

Example 1

easy

A photo is 4 in × 6 in. It is enlarged with scale factor

k = 3

. What are the new dimensions and new area?

Example 2

hard

Two similar pyramids have heights 4 m and 10 m. If the smaller pyramid has volume

32

m³, what is the volume of the larger?

Example 3

easy

If you scale a shape's lengths by a factor of

3

, by what factor does its area change?

Example 4

easy

Scale a solid's lengths by

2

. By what factor does its volume change?

Example 5

easy

Scale lengths by

5

. By what factor does length itself change?

Example 6

easy

A photo is enlarged by a scale factor of

4

. By what factor does its area increase?

Example 7

easy

A scale factor of

\tfrac{1}{2}

shrinks a solid. By what factor does its volume change?

Example 8

easy

Lengths scale by

k

. Match each measure to its exponent: length, area, volume.

Example 9

easy

A square of area

5

is scaled so its sides triple. Find the new area.

Example 10

easy

A cube of volume

2

has its sides doubled. Find the new volume.

Example 11

medium

Two similar solids have a length ratio of

2:3

. Find the ratio of their volumes.

Example 12

medium

Two similar figures have areas

16

and

25

. Find the ratio of their perimeters.

Example 13

medium

A model car is built at

\tfrac{1}{20}

scale. The real car has

4\,\text{m}^2

of paint surface. How much surface does the model have?

Example 14

medium

A statue is scaled up by factor

4

. Its original weight (proportional to volume) was

5

kg. Find the new weight.

Example 15

medium

Two similar cans have volumes

54

and

128

. Find the ratio of their heights.

Example 16

medium

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

Example 17

medium

If a shape's area increases by a factor of

9

, by what factor did its lengths scale?

Example 18

medium

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

Example 19

challenge

Two similar prisms have surface areas

50

and

200

and the larger has volume

640

. Find the smaller prism's volume.

Example 20

challenge

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

Example 21

challenge

A cone is filled with water to

\tfrac{1}{3}

of its height. What fraction of the cone's volume is filled?

Example 22

challenge

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

Example 23

easy

A rectangle has area

24

. All lengths are scaled by

5

. Find the new area.

Example 24

easy

A cube has volume

1\text{ cm}^3

. Scale lengths by

4

. Find the new volume.

Example 25

easy

A triangle's sides are doubled. By what factor does its perimeter change?

Example 26

easy

A sphere has surface area

A

. Scale radius by

3

. Find the new surface area.

Example 27

easy

If a figure's area increases by

36

times, by what factor did the lengths scale?

Example 28

easy

If a solid's volume grows

125

-fold, by what factor did each side scale?

Example 29

medium

Two similar pentagons have areas

48

and

108

. Find the ratio of corresponding sides.

Example 30

medium

Two similar cones have heights

4

and

7

. Find the ratio of their volumes.

Example 31

medium

A sphere's surface area grows from

100

400

^2

. By what factor did its radius scale, and what is the volume factor?

Example 32

medium

Two similar boxes have surface areas

50

and

200

. The larger holds

640

^3

. How much does the smaller hold?

Example 33

medium

A 16-inch pizza (

16

in diameter) costs

\$16

and an

8

-inch pizza costs

\$5

. Compare price per unit area; which is the better deal?

Example 34

medium

1:10

scale model car weighs

2

kg. If made of the same material, what would the real car weigh?

Example 35

medium

If volume scales by factor

216

, by what factor does surface area scale?

Example 36

hard

An object's surface area is

A

and its volume is

V

. Scale lengths by

k

. Show that the ratio

V/A

scales by

k

(linearly).

Example 37

hard

A cone of height

H

is filled with water to height

h

. What fraction of the cone's volume is the water, in terms of

h/H

Example 38

hard

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

Example 39

hard

Two similar pyramids have volumes

54

and

128

. Find the ratio of their heights and the ratio of their surface areas.

Example 40

hard

A photograph is enlarged from

6\text{ in} \times 4\text{ in}

15\text{ in} \times 10\text{ in}

. By what factor does its area change?

Example 41

hard

Strength of a bone scales with cross-sectional area (

k^2

). Body weight scales with volume (

k^3

). Show that for a uniformly scaled animal, relative strength (strength/weight) declines as

1/k

Example 42

challenge

Two similar solid balls of the same material have masses

27

kg and

125

kg. Find the ratio of their surface areas.

Example 43

challenge

A cone is partially filled with water so the water occupies

\tfrac{1}{8}

of the cone's volume. What fraction of the cone's height does the water reach?

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Related Concepts

Background Knowledge

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

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