How to solve SAT area and volume questions

Area and volume questions on the Digital SAT ask you to find the area of flat shapes, the volume or surface area of solids, or a missing dimension when you know one of those measures. Some questions skip formulas and test how area and volume change when a shape is scaled. Geometry and Trigonometry is about 15% of the Math section, or 5 to 7 of the 44 questions. Those questions are split across four skills, so expect 1 or 2 area and volume questions per test.

The reference sheet in the testing app gives the formulas for the area of a circle, rectangle, and triangle, and the volume of a rectangular prism, cylinder, sphere, cone, and pyramid. You still need to know when to use each one and how to work backward from them.

What the questions look like

Direct formula use. You find a volume or area from given dimensions, or find a radius, height, or side length from a given volume or area. Answers are often left in terms of $\pi$.

Composite figures. A shape is made of simpler pieces, like a rectangle with a semicircle on one end, or a square with a circle cut out. You add or subtract the pieces.

Scaling and comparison. Two similar solids are compared, or one dimension of a shape changes, and you find the new area, volume, or the ratio between them.

Step-by-step method

1. Sketch and label. Write every given dimension on a quick drawing. Note whether you were given a radius or a diameter.
2. Choose the formula. The volume formulas you will use most:
   • Rectangular prism: $V = \ell wh$
   • Cylinder: $V = \pi r^2 h$
   • Cone: $V = \frac{1}{3}\pi r^2 h$
   • Pyramid: $V = \frac{1}{3}\ell wh$
   • Sphere: $V = \frac{4}{3}\pi r^3$
3. For surface area, add the areas of all faces. A rectangular prism has surface area $2(\ell w + \ell h + wh)$.
4. For composite figures, split the shape into pieces whose areas you know, then add or subtract.
5. For scaling, if every length is multiplied by $k$, areas are multiplied by $k^2$ and volumes by $k^3$. To go from a volume ratio back to a length ratio, take the cube root.
6. Solve for the unknown. Substitute what you know into the formula and solve the resulting equation.

Worked example

Example 1. A right circular cylinder and a right circular cone both have a radius of 6 inches. The cylinder has a height of 10 inches. The cone has the same volume as the cylinder. What is the height of the cone?

Find the cylinder's volume:

$$V = \pi (6)^2 (10) = 360\pi$$

Set the cone's volume equal to $360\pi$:

$$\frac{1}{3}\pi (6)^2 h = 360\pi$$
$$12\pi h = 360\pi$$
$$h = 30$$

The cone is 30 inches tall. This matches a general fact: a cone with the same radius as a cylinder needs 3 times the height to hold the same volume.

Example 2. A rectangular prism has a volume of 60 cubic centimeters and a surface area of 94 square centimeters. A second prism is similar to the first and has a volume of 1,620 cubic centimeters. What is the surface area of the second prism?

The volume ratio is

$$\frac{1620}{60} = 27$$

Volumes scale by $k^3$, so $k^3 = 27$ and $k = 3$. Every length of the second prism is 3 times as long. Surface areas scale by $k^2 = 9$:

$$94 \times 9 = 846 \text{ square centimeters}$$

You can confirm with actual dimensions. A $3 \times 4 \times 5$ prism has volume 60 and surface area $2(12 + 15 + 20) = 94$. Scaling by 3 gives $9 \times 12 \times 15$, with volume 1,620 and surface area $2(108 + 180 + 135) = 846$.

Common traps

• Using the diameter as the radius. If a problem says a sphere has a diameter of 10, use $r = 5$. Plugging in 10 makes the volume 8 times too large.
• Scaling area or volume by $k$. Doubling every edge of a cube multiplies its volume by 8, not 2. In Example 2, multiplying 94 by 27 or by 3 gives common wrong answers.
• Forgetting the $\frac{1}{3}$ for cones and pyramids. A cone holds one third of the cylinder with the same base and height. Check the reference sheet if you are unsure.
• Mixing units. If the radius is in inches and the height is in feet, convert first. For cubic units, the conversion factor is cubed: 1 cubic foot is $12^3 = 1{,}728$ cubic inches.
• Confusing surface area with volume. Surface area is in square units and counts the outside of the solid. Volume is in cubic units and measures what fits inside. Read which one the question asks for.

Desmos tip

When the answer choices are written in terms of $\pi$, compute the numerical answer in Desmos and then divide by $\pi$ to see the coefficient. For example, typing (1/3)*pi*6^2*30 returns about 1130.97, and dividing that by pi returns 360, matching a choice of $360\pi$. To solve for a missing dimension, type the formula as an equation in $x$, such as (1/3)*pi*36*x = 360*pi, and read the solution from the vertical line Desmos draws.