Scale the right dimension.

For similar shapes, corresponding lengths share a multiplier k. Areas multiply by $k^{2}$ because two dimensions change; volumes multiply by $k^{3}$ because three dimensions change. Angles stay equal.

A diagram does not need to be drawn to scale if its measurements are labelled. Write the formula before substituting, and convert all lengths to compatible units. The area of a circle uses $r^{2}$ , while its circumference uses r.

INTERACTIVE MODEL

Try it. Watch it change.

Area / square units8

\sqrt{8}

= 2

\sqrt{2}

≈ 2.8284

A square’s side length is the positive square root of its area. Exact surd form avoids rounding.

Explore: Increase square area from 4 to 16. Observe how the side length changes compared with the area.

Scale factors

\begin{aligned}\text{length}&:\; k\\\text{area}&:\; k^2\\\text{volume}&:\; k^3\end{aligned}

WORKED EXAMPLE

A similar solid has twice every length of another solid. How do surface area and volume change?

The length factor is 2.
Surface area factor = $2^{2}$ = 4.
Volume factor = $2^{3}$ = 8.

Assumed knowledge

Areas of rectangles, triangles and circles; volume units.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Distinguish length, area and volume scale factors.
Use units and formulas to solve measurement problems.

Year 10 preparation ↗

READY TO TRY IT?

Scale the right dimension.

A similar solid has twice every length of another solid. How do surface area and volume change?

Put the idea to work.