18.6 Similarity: areas and volumes — Learn

Prerequisites: similarity and scale factors, area of rectangles, and volume of cuboids.

Objectives: relate length, area and volume scale factors for similar shapes and solids.

When a shape is enlarged by length scale factor $k$, each length is multiplied by $k$. An area uses two lengths, so its scale factor is $k^2$. A volume uses three lengths, so its scale factor is $k^3$.

Example. A similar shape has length scale factor $3$.

Area scale factor $=3^2$. $3^2=9$. An area of $7$ cm$^2$ becomes $7\times9=63$ cm$^2$.

Example. A similar solid has volume scale factor $125$.

Length scale factor $=\sqrt[3]{125}$. $\sqrt[3]{125}=5$. Each matching length is multiplied by $5$.

Watch out. Do not use the length scale factor directly on area or volume. Check the power: square for area, cube for volume.

Progress check

The mural's lengths double. Predict the area multiplier and the volume multiplier. Then explain why area and volume use different powers.

You should now be able to

• find area and volume scale factors from a length scale factor;
• find a length scale factor from an area or volume ratio;
• solve problems involving similar areas and solids;
• explain why area is squared and volume is cubed.