Multiply the surface area of A by 9: $$ 120 \times 9=1080 $$ 4. The total surface area of B is 1080 cm². If a length doubles, the surface area does not just double. It is multiplied by 4, because the area scale factor is $2^2$. ## 3. Volume: cube the scale factor Volume is based on three lengths: length by width by height. So if all lengths are multiplied by $k$, all volumes are multiplied by $k^3$. If the length scale factor is $k$, then the volume scale factor is $k^3$. Two similar cylinders have lengths 5 cm and 8 cm. The smaller cylinder has volume 250 cm³. Find the volume of the larger cylinder. 1. Use the corresponding lengths to find the scale factor from smaller to larger: $$ k=\frac{8}{5} $$ 2. Cube the scale factor to get the volume scale factor: $$ \left(\frac{8}{5}\right)^3=\frac{512}{125} $$ 3. Multiply the smaller volume by the volume scale factor: $$ 250 \times \frac{512}{125}=1024 $$ 4. The volume of the larger cylinder is 1024 cm³. If you are going from a larger shape to a smaller shape, you can either use a fractional scale factor or divide by the volume scale factor. Both methods work, as long as your direction is clear. ## 4. Working backwards from surface area Sometimes you are given two surface areas, but asked for a volume. In that case, you must work in stages: 1. Find the area scale factor. 2. Square root it to get the length scale factor. 3. Cube the length scale factor to get the volume scale factor. Two similar solid shapes X and Y have total surface areas 180 cm² and 320 cm². The volume of X is 540 cm³. Find the volume of Y. 1. Find the area scale factor from X to Y: $$ \frac{320}{180}=\frac{16}{9} $$ 2. Square root the area scale factor to get the length scale factor: $$ k=\sqrt{\frac{16}{9}}=\frac{4}{3} $$ 3. Cube the length scale factor to get the volume scale factor: $$ \left(\frac{4}{3}\right)^3=\frac{64}{27} $$ 4. Multiply the volume of X by the volume scale factor: $$ 540 \times \frac{64}{27}=1280 $$ 5. The volume of Y is 1280 cm³. Do not cube the area scale factor. The area scale factor already has a square built into it, so square-root it first. ## 5. Frustums: subtract cone volumes A **frustum** is the solid left when the top of a cone or pyramid has been cut off parallel to its base. For cone questions, the frustum volume is found by subtracting the small cone volume from the large cone volume. For a cone: $$ V=\frac{1}{3}\pi r^2h $$ Here, $r$ is the radius and $h$ is the height. The diameter is the full width across a circle through the centre, so the radius is half the diameter. An answer “in terms of $\pi$” means leave $\pi$ in your answer rather than using a decimal approximation. A large cone has height 9 cm and base diameter 24 cm. A smaller similar cone at the top has height 3 cm and base diameter 8 cm. Find the volume of the frustum in terms of $\pi$. 1. Convert the diameters into radii: $$ \begin{aligned} r_{\text{large}}&=12\\ r_{\text{small}}&=4 \end{aligned} $$ 2. Find the volume of the large cone: $$ V_{\text{large}}=\frac{1}{3}\pi \times 12^2 \times 9=432\pi $$ 3. Find the volume of the small cone: $$ V_{\text{small}}=\frac{1}{3}\pi \times 4^2 \times 3=16\pi $$ 4. Subtract to get the frustum volume: $$ V_{\text{frustum}}=432\pi-16\pi=416\pi $$ 5. The volume of the frustum is $416\pi$ cm³. The cone formula uses radius, not diameter. If a question gives a diameter,