Area and volume on the SAT

Example 1: cylinder volume

Problem A cylinder has radius 3 and height 10. What is its volume?

1. Use $V = \pi r^2 h$.
   $$V = \pi (3)^2 (10) = 90\pi$$

Answer $90\pi$, about 282.7 cubic units

Example 2: work backward to a length

Problem A rectangular box has a volume of 360 cubic centimeters, a length of 12 cm and a width of 5 cm. What is its height?

1. Write the volume formula with what you know.
   $$12 \cdot 5 \cdot h = 360$$
2. Simplify.
   $$60h = 360$$
3. Divide by 60.
   $$h = 6$$

Answer 6 cm

Example 3: scaling

Problem Every edge of a cube is tripled. By what factor does the volume change?

1. Volume scales by the cube of the length factor.
   $$3^3 = 27$$
2. Check with a 1 by 1 by 1 cube: tripled, it becomes 3 by 3 by 3, and $27 \div 1 = 27$.

Answer The volume is multiplied by 27.

Example 4 (SAT-hard): equal volumes

Problem A cone and a cylinder both have radius 4. The cylinder's height is 9. The two solids have the same volume. What is the height of the cone?

1. Cylinder volume.
   $$\pi (4)^2 (9) = 144\pi$$
2. Cone volume with height h.
   $$\frac{1}{3}\pi (4)^2 h = \frac{16}{3}\pi h$$
3. Set them equal and solve.
   $$\frac{16}{3}\pi h = 144\pi$$
4. A cone holds one third of a cylinder with the same base and height, so it needs 3 times the height: $3 \times 9 = 27$.

Answer 27