Similar solids, like similar polygons in Similarity Through Transformations, have every length multiplied by the same scale factor $k$. Areas and volumes change by more, because they involve two or three lengths multiplied together. The formulas from Volume of Pyramids, Cones and Spheres show it.
Why it works
Multiply every length of a box by $k$. Its surface area is made of products of two lengths, like $lw$, each of which becomes $(kl)(kw) = k^2 lw$: so area is multiplied by $k^2$. Its volume $lwh$ becomes $(kl)(kw)(kh) = k^3 lwh$. The same is true for any solid, since every area and volume formula multiplies two or three lengths.
Key idea
If lengths are scaled by $k$, areas (including surface area) are scaled by $k^2$ and volumes by $k^3$.
Worked example: Area and volume
A model car is built at a scale of $1 : 20$. The model's surface area is $150$ cm$^2$ and its volume is $400$ cm$^3$. Estimate the real car's surface area and volume.
- $k = 20$. Area: $20^2 \cdot 150 = 400 \cdot 150 = 60{,}000$ cm$^2$, which is $6$ m$^2$.
- Volume: $20^3 \cdot 400 = 8000 \cdot 400 = 3{,}200{,}000$ cm$^3$, which is $3.2$ m$^3$.
Answer: About $60{,}000$ cm$^2$ and $3{,}200{,}000$ cm$^3$
Worked example: Working backward
Two similar cylinders have volumes $54\pi$ and $250\pi$. The smaller one is $6$ cm tall. How tall is the larger one?
- Volume ratio: $\frac{250}{54} = \frac{125}{27} = k^3$, so $k = \frac{5}{3}$.
- Height: $6 \cdot \frac{5}{3} = 10$ cm.
Answer: $10$ cm
Worked example: Surface area from volume
Two similar solids have volumes in the ratio $8 : 27$. The smaller one has a surface area of $36$ in$^2$. Find the larger surface area.
- $k^3 = \frac{27}{8}$, so $k = \frac{3}{2}$, and $k^2 = \frac{9}{4}$.
- $36 \cdot \frac{9}{4} = 81$ in$^2$.
Answer: $81$ in$^2$
Common mistake
Doubling the volume when every length is doubled. If a box's length, width and height all double, its volume is $2 \cdot 2 \cdot 2 = 8$ times as large, and its surface area $4$ times as large. Check with unit cubes: a $2 \times 2 \times 2$ cube is made of 8 small ones.