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Area, perimeter, volume, and surface area — SHSAT practice questions

Area, perimeter, volume, and surface area on the SHSAT Mathematics section covers two- and three-dimensional measurement: finding a missing length from a given area or perimeter, converting among related formulas, and counting unit cubes inside a solid. Students must recall circle formulas, scale rectangle sides from a given ratio and perimeter, and distinguish interior cubes from those on an outside face. Questions are typically multiple choice. Answer choices are numbers or expressions with pi, and distractors often come from mixing radius with diameter, using area when perimeter is needed, or counting every cube instead of only those that do not touch the exterior.

  • Section: Mathematics 57 questions on the paper
  • 141 practice questions in the app

Sample questions

Question 1

The side lengths of a rectangle are in the ratio 5:85:8. If the perimeter of the rectangle is 78, what is the area of the rectangle?

  1. A117
  2. B195
  3. C312
  4. D360
Show answer

Answer: D — 360

The perimeter is 2(5k+8k)=26k2(5k+8k)=26k. Since 26k=7826k=78, k=3k=3. The side lengths are 15 and 24, so the area is 15⋅24=36015\cdot24=360.

Question 2

A rectangular prism is built from unit cubes and has dimensions 4 units by 5 units by 6 units. How many of the unit cubes do not touch any outside face of the prism?

  1. A12
  2. B18
  3. C24
  4. D36
Show answer

Answer: C — 24

Removing the outside layer from each dimension leaves an interior block measuring (4−2)(4-2) by (5−2)(5-2) by (6−2)(6-2) unit cubes. That is 2⋅3⋅4=242\cdot3\cdot4=24 cubes that do not touch any outside face.

Question 3

A circular window has an area of 121π121\pi square inches. What is the circumference of the window in inches?

  1. A11π11\pi
  2. B22π22\pi
  3. C121π121\pi
  4. D242π242\pi
Show answer

Answer: B — 22π22\pi

From πr2=121π\pi r^2=121\pi, the radius is 11 inches. Therefore, the circumference is 2π(11)=22π2\pi(11)=22\pi inches.

Practice 141 Area, perimeter, volume, and surface area questions in the app

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