Scaling Laws Examples in Math

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

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

Concept Recap

Relationships describing how a quantity changes when the size or scale of a system is multiplied by a factor, often expressed as power laws.

When you double the length of a cube, its volume grows by $2^3 = 8$ . Scaling laws reveal how fast quantities grow — they often explain why small and large things behave so differently.

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: Scaling laws describe how a quantity grows as a power of size when you multiply the scale by a factor.

Common stuck point: The procedure for scaling laws is the easy part; the trap is scaling area or volume by the length factor. Asking "When I multiply length by $k$ , does the quantity multiply by $k$ , $k^2$ , $k^3$ , or another power?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: When I multiply length by $k$ , does the quantity multiply by $k$ , $k^2$ , $k^3$ , or another power?

Worked Examples

Example 1

easy

If you scale a square's side length by a factor of 3, by what factor does the area change? State the general scaling law.

Answer

\text{Area scales by } k^2 = 9 \text{ when lengths scale by } k=3

First step

Original square: side

s

, area

s^2

Full solution

2
Scaled square: side $3s$ , area $(3s)^2 = 9s^2$ .
3
Area scales by $3^2 = 9$ .
4
General law: scaling all lengths by factor $k$ scales area by $k^2$ .

Scaling laws describe how quantities change when a single scale factor is applied. Area is a 2-dimensional quantity, so it scales as

k^2

. This law is universal for any 2D shape, not just squares.

Example 2

medium

A sphere of radius

r

has volume

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

. If the radius doubles, by what factor does the volume increase? Generalise to a scaling factor

k

Example 3

medium

Two similar cylinders have heights in ratio

2:5

. Find the ratios of their surface areas and volumes.

Example 4

medium

Why do small mammals lose body heat faster than large ones? Use surface-area-to-volume scaling to explain.

Example 5

medium

Two similar triangles have areas

9

and

25

. What is the ratio of their corresponding sides?

Example 6

hard

A model bridge supports a load of

10

kg. If the real bridge has linear scale

100\times

the model and uses the same materials, the load it can support scales as cross-sectional area (

L^2

) while its weight scales as volume (

L^3

). Find the ratio of supported load to its own weight, compared to the model.

Example 7

hard

For a body falling under gravity, distance

\propto t^2

. If a stone falls

5

m in

1

s, how far does it fall in

3

s (ignoring air resistance)?

Example 8

hard

Two similar containers have heights

h_1=10

and

h_2=15

. The smaller holds

40

L. Find the capacity of the larger.

Example 9

challenge

A 2D fractal has Hausdorff dimension

D

between

1

and

2

. If you scale linear size by factor

k

, its 'mass' (or measure) scales by

k^D

. For the Sierpinski triangle (

D=\log_2 3

), find the mass factor when linear size doubles.

Practice Problems

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

Example 1

easy

If the radius of a circle doubles, by what factor does the circumference change? What about the area?

Example 2

medium

A population model gives

P(t) = P_0 e^{rt}

. If

r

is doubled (growth rate doubles), by what factor does

P(T)

easy

If a square's side is tripled, by what factor does its area change?

Example 24

easy

Two similar rectangles have a length ratio of

1:5

. What is the ratio of their areas?

Example 25

easy

y\propto \sqrt{x}

and

x

is multiplied by

9

, by what factor does

y

change?

Example 26

easy

A square's side increases from

4

12

. By what factor does its area increase?

Example 27

medium

A scale model of a building has linear scale

1:50

. If the model's roof has area

0.2

^2

, find the actual roof area.

Example 28

medium

A scale model car has volume

0.001

^3

at scale

1:20

. Find the actual volume.

Example 29

medium

Kepler's third law states

T^2\propto r^3

for planetary orbits. If a planet's orbital radius doubles, by what factor does its period change?

Example 30

medium

Two similar cones have volumes

8

^3

and

125

^3

. Find the ratio of their heights.

Example 31

hard

A 6-inch pizza costs \$8 and a 12-inch pizza costs \$24. Which is the better deal per square inch?

Example 32

hard

y\propto x^{-2}

(inverse square law) and

x

is doubled, by what factor does

y

change?

Example 33

hard

A computer's runtime is

O(n^2)

. If input size doubles, by what factor does runtime change? If it goes from

n

4n

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Related Concepts

Dimensional Reasoning Proportionality

Background Knowledge

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

dimensional reasoningproportionality