Use scale factors to reason about changes in length and area (and volume when relevant). | Area and volume | SAT Math

Scale factors change length by k, area by k^2, and volume by k^3.

Core Idea

When every length in a figure is multiplied by a scale factor $𝑘$ , area is multiplied by $𝑘 2$ and volume by $𝑘 3$ .

Understanding

Rule: A scale factor changes length, area, and volume at different rates.

Length and perimeter scale by $𝑘$ .
Area scales by $𝑘 2$ .
Volume scales by $𝑘 3$ .

Ask whether the whole figure scaled or only one dimension did.

Step by Step

Find the scale factor $𝑘$ — the ratio of corresponding lengths.
For area, multiply by $𝑘 2$ .
For volume, multiply by $𝑘 3$ .
If only some dimensions change, apply the factor only to those dimensions in the formula.

Misconceptions

Applying $𝑘$ directly to area (instead of $𝑘 2$ ) or to volume (instead of $𝑘 3$ ).
Assuming all dimensions scale when only one does (e.g., doubling height but not radius).
Confusing the scale factor direction — using $𝑘$ when $1 𝑘$ is needed.

Question

Worked Example

Two similar cylinders have radii 2 cm and 6 cm. If the smaller cylinder has a volume of $20 𝜋$ cubic centimeters, what is the volume of the larger cylinder, in cubic centimeters?

Select an answer to see the explanation