PSAT 8/9 Math: Area and Volume
Area and volume questions on the PSAT 8/9 test your comfort applying formulas, and rearranging them to solve for a missing piece. This free, complete lesson covers area and perimeter of rectangles, triangle area and missing sides, volume of cubes and rectangular prisms, and solving for a missing dimension. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Area and Perimeter of Rectangles
A rectangle's area is length × width, while its perimeter is the distance around it: 2(length + width). Keep track of which one a question asks for, since area and perimeter measure very different things.
Worked Example: Find the area of a rectangle with length 9 inches and width 5 inches.
Area = 9 × 5 = 45 square inches
1. Find the area of a rectangle with length 9 inches and width 5 inches. (see worked example above)
Explanation: Area = length × width = 9 × 5 = 45 square inches.
2. Find the perimeter of a rectangle with length 15 and width 4.
Explanation: Perimeter = 2(length + width) = 2(15 + 4) = 2(19) = 38.
3. A square has an area of 81 square inches. Find its side length.
Explanation: A square's area equals side², so side = √81 = 9 inches.
2. Triangle Area and Missing Sides
A triangle's area is 12 × base × height. If you know a triangle's perimeter and two of its three sides, subtract to find the missing side.
Worked Example: Find the area of a triangle with base 10 feet and height 6 feet.
Area = (1/2) × 10 × 6 = 30 square feet
1. Find the area of a triangle with base 10 feet and height 6 feet. (see worked example above)
Explanation: Area = (1/2) × base × height = (1/2) × 10 × 6 = 30 square feet.
2. A triangle has two sides measuring 5 cm and 7 cm, and a perimeter of 20 cm. Find the third side.
Explanation: 20 − 5 − 7 = 8 cm.
3. A triangle has a base of 14 meters and an area of 42 square meters. Find its height.
Explanation: 42 = (1/2) × 14 × h → 42 = 7h → h = 6 meters.
3. Volume of Cubes and Rectangular Prisms
A cube's volume is side³ (side × side × side). A rectangular prism's volume is V = lwh (length × width × height).
Worked Example: Find the volume of a cube with side length 4 inches.
Volume = 4³ = 4 × 4 × 4 = 64 cubic inches
1. Find the volume of a cube with side length 4 inches. (see worked example above)
Explanation: Volume of a cube = side³ = 4³ = 4 × 4 × 4 = 64 cubic inches.
2. A rectangular prism has length 6 cm, width 2 cm, and height 3 cm. Use V = lwh to find its volume.
Explanation: V = lwh = 6 × 2 × 3 = 36 cubic cm.
3. A cube has a volume of 125 cubic feet. Find its side length.
Explanation: Since volume = side³, side = ∛125 = 5 feet.
4. Solving for a Missing Dimension
When a volume or area and some, but not all, dimensions are known, substitute what you know into the formula and solve algebraically for the missing dimension.
Worked Example: A box has a volume of 96 cubic inches, a length of 8 inches, and a width of 3 inches. Find its height.
Substitute into V = lwh: 96 = 8 × 3 × h.
96 = 24h.
h = 4 inches
1. A box has a volume of 96 cubic inches, a length of 8 inches, and a width of 3 inches. Find its height. (see worked example above)
Explanation: 96 = 8 × 3 × h → 96 = 24h → h = 4 inches.
2. A rectangle has an area of 72 square meters and a width of 8 meters. Find its length.
Explanation: Length = Area / width = 72 / 8 = 9 meters.
3. A rectangular prism has a volume of 150 cubic cm, a length of 10 cm, and a height of 3 cm. Find its width.
Explanation: 150 = 10 × w × 3 → 150 = 30w → w = 5 cm.
Common Mistakes to Avoid
- Confusing area and perimeter, area is a product (measured in square units), while perimeter is a sum (measured in linear units).
- Forgetting the 1/2 in the triangle area formula, a triangle's area is always half of what a rectangle with the same base and height would have.
- Mixing up which formula applies to a cube versus a general rectangular prism, a cube is a special case where all three dimensions are equal (side³), but a general prism needs V = lwh with three different dimensions.
- Solving for the wrong variable when rearranging a formula, always isolate the exact dimension the question asks for.
- Forgetting to label units correctly, area answers should be in square units, and volume answers should be in cubic units.
Frequently Asked Questions
Do I need to memorize area and volume formulas for the PSAT 8/9?
The most commonly used formulas are usually provided in a reference box at the start of the Math section. Still, being comfortable applying and rearranging them quickly will save valuable time.
How is a cube's volume formula related to a rectangular prism's?
A cube is simply a rectangular prism where length, width, and height are all equal. So instead of multiplying three different numbers (V = lwh), you can just cube one number (V = side³).
Where can I practice more problems like these?
The Area and Volume quizzes in the PSAT 8/9 Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT 8/9. You can also browse the full PSAT 8/9 Math Question Bank.