PSAT/NMSQT Math: Area, Perimeter, and Volume
Area, perimeter, and volume questions on the PSAT/NMSQT test how well you can apply geometric formulas, rearrange them to solve for a missing piece, and scale them for similar figures. This free, complete lesson covers perimeter and missing side lengths, area of rectangles and triangles, volume of prisms and cones, and similar figures and scale factors. 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. Perimeter and Missing Side Lengths
The perimeter of a polygon is the sum of all its side lengths. For a square with side s, perimeter = 4s. If the perimeter and some side lengths are known, subtract the known sides from the total perimeter to find a missing side.
Worked Example: A triangle has a perimeter of 28 inches. Two of its sides measure 9 inches and 8 inches. Find the third side.
28 − 9 − 8 = 11 inches
1. A square has a side length of 27. What is its perimeter?
Explanation: Perimeter of a square = 4 × side = 4 × 27 = 108.
2. A triangle has a perimeter of 28 inches. Two of its sides measure 9 inches and 8 inches. Find the third side. (see worked example above)
Explanation: 28 − 9 − 8 = 11 inches.
3. An equilateral triangle has a perimeter of 45 units. What is the length of one side?
Explanation: An equilateral triangle has 3 equal sides, so one side = 45 / 3 = 15 units.
2. Area of Rectangles and Triangles
Area of a rectangle = length × width. Area of a triangle = 12 × base × height. Some problems describe one dimension in terms of another, so you may need to write and solve an expression before applying the formula.
Worked Example: A rectangle has a width of 9 cm, and its length is 35 cm more than its width. Find its area.
Length = 9 + 35 = 44 cm.
Area = 9 × 44 = 396 cm²
1. A rectangle has a width of 9 cm, and its length is 35 cm more than its width. Find its area. (see worked example above)
Explanation: Length = 9 + 35 = 44 cm. Area = 9 × 44 = 396 cm².
2. A triangle has a base of 36 cm and a height of 50 cm. Find its area.
Explanation: Area = (1/2) × base × height = (1/2) × 36 × 50 = 900 cm².
3. A rectangular garden has an area of 120 square feet and a length of 15 feet. Find its width.
Explanation: Width = Area / length = 120 / 15 = 8 feet.
3. Volume of Prisms and Cones
Volume of a rectangular prism = length × width × height. Volume of any prism = base area × height. Volume of a cone = 13 × π × r² × height. If the volume is known and one dimension is missing, rearrange the formula to solve for that dimension.
Worked Example: A rectangular prism has a volume of 2,700 cubic inches, a length of 15 inches, and a width of 12 inches. Find its height.
h = 2,700 / (15 × 12) = 2,700 / 180 = 15 inches
1. A rectangular prism has a volume of 2,700 cubic inches, a length of 15 inches, and a width of 12 inches. Find its height. (see worked example above)
Explanation: Volume = l × w × h, so h = 2,700 / (15 × 12) = 2,700 / 180 = 15 inches.
2. A cone has a radius of 4 units and a height of 9 units. Find its volume. (Use π ≈ 3.14; round to the nearest whole number.)
Explanation: Volume = (1/3) × π × r² × h = (1/3) × 3.14 × 16 × 9 = (1/3) × 452.16 ≈ 151 cubic units.
3. A hexagonal prism has a base area of 30 square inches and a height of 8 inches. Find its volume.
Explanation: Volume of any prism = base area × height = 30 × 8 = 240 cubic inches.
4. Similar Figures and Scale Factors
For similar figures, corresponding sides are proportional. If two triangles are similar with a scale factor of k between corresponding sides, then their perimeters also scale by k (perimeter is a length-based measurement, so it scales the same way individual sides do).
Worked Example: Two triangles are similar, and corresponding sides of the larger triangle are 3 times the corresponding sides of the smaller triangle. If the smaller triangle's perimeter is 18, find the larger triangle's perimeter.
Larger perimeter = 3 × 18 = 54
1. Two triangles are similar, and corresponding sides of the larger triangle are 3 times the corresponding sides of the smaller triangle. If the smaller triangle's perimeter is 18, find the larger triangle's perimeter. (see worked example above)
Explanation: Perimeter scales by the same factor as the sides: 3 × 18 = 54.
2. Two similar triangles have corresponding sides of 5 and 20. If the smaller triangle has a perimeter of 24, find the larger triangle's perimeter.
Explanation: Scale factor = 20/5 = 4. Larger perimeter = 4 × 24 = 96.
3. A model of a building is built at a scale where 1 inch represents 20 feet. If the model is 8 inches tall, how tall is the actual building?
Explanation: 8 inches × 20 feet/inch = 160 feet.
Common Mistakes to Avoid
- Mixing up area and perimeter formulas. Perimeter is a sum of side lengths, area is a product, and volume is a triple product, keep track of which units (linear, square, or cubic) each answer should have.
- Forgetting the 1/2 in the triangle area formula or the 1/3 in the cone volume formula, both come from comparing these shapes to a related shape, a rectangle or a cylinder, that has no such fraction.
- Solving for the wrong dimension when rearranging a volume or area formula, always isolate the specific unknown the question asks for, not a different variable in the formula.
- Assuming perimeters of similar figures share the same ratio as their areas. Perimeter scales linearly with the scale factor, while area scales with the scale factor squared, these are two different relationships.
- Not reading a dimension description carefully, such as "length exceeds width by 35," which requires adding to the width rather than using 35 as the length directly.
Frequently Asked Questions
Do I need to memorize every area and volume formula for the PSAT/NMSQT?
The most commonly needed formulas, including area of a rectangle, triangle, and circle, and volume of a rectangular prism, cylinder, and cone, are provided in a reference box at the start of the Math section. Still, being comfortable using them quickly, and rearranging them to solve for a missing variable, saves valuable time.
How is volume of a prism related to area?
Any prism's volume equals its base area multiplied by its height, so finding a prism's volume is really a two-step process: find the area of the base shape first (a rectangle, triangle, or hexagon, for instance), then multiply by the height.
Where can I practice more problems like these?
The Area, Perimeter, and Volume quiz in the PSAT/NMSQT Math Question Bank includes 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/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.