ACT · Math · Integrating Essential Skills

Area, perimeter, and volume

Area, perimeter, and volume is part of Integrating Essential Skills in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include perimeter and fencing problems, area of a composite figure, volume of a prism or cylinder, volume of a cone, sphere and or pyramid.

Common traps: including the open side; using area instead of perimeter; forgetting to multiply by the cost per unit.

Sample questions

Question 1
A banner for a balloon festival is a rectangle 6 m wide and 5 m tall with an isosceles triangle added on top. The triangle's base is the 6 m top edge of the rectangle, its two equal sides are each 5 m, and its height is 4 m. What is the total area of the banner, in square meters?
- A.
  30
- B.
  42
- C.
  46
- D.
  54
Answer: B. 42
The rectangle has area 6 × 5 = 30 and the triangle has area (1/2)(6)(4) = 12, so the banner has area 42 m². Choice A (30) leaves out the triangle; C (46) adds the triangle's perimeter, 6 + 5 + 5, instead of its area; D (54) forgets the factor 1/2 in the triangle's area.
Question 2
A right circular cone has a radius of 3 inches and a height of 4 inches. What is its volume, in cubic inches?
- A.
  12π
- B.
  4π
- C.
  36π
- D.
  24π
Answer: A. 12π
V = (1/3)πr²h = (1/3)π(3)²(4) = 12π.
Question 3
A farmer in Kenya builds a rectangular goat pen with an area of 180 square meters and a width of 9 m. Fencing costs KES 1,200 per meter. One 2 m section of the perimeter is a gate, which replaces 2 m of fencing and costs a flat KES 6,000. What is the total cost of the fencing and the gate, in shillings?
- A.
  64,800
- B.
  67,200
- C.
  69,600
- D.
  73,200
Answer: D. 73,200
The length is 180 ÷ 9 = 20 m, so the perimeter is 58 m. Fencing is 56 m at KES 1,200 = 67,200, plus the 6,000 gate: 73,200. Choice A leaves out one 9 m side; B omits the cost of the gate; C charges for the whole perimeter, with no gate at all.
Question 4
A vendor at a street market in Mexico City lines the inside of an open-top cubical storage bin with wax paper. Each edge of the bin is 40 cm long, and the paper covers the four walls and the bottom. How many square centimeters of paper are needed?
- A.
  8,000
- B.
  9,600
- C.
  16,000
- D.
  64,000
Answer: A. 8,000
Each face is 40 × 40 = 1,600 cm², and there are 5 faces to cover (four walls and the bottom), so 5 × 1,600 = 8,000 cm². Choice B counts a sixth face for the open top; C covers the inside and the outside of five faces; D is the volume, 40³, in cubic centimeters.
Question 5
Square ABCD has side length 10 cm. Quarter circle P is centered at A with radius 10 cm and passes through B and D. Quarter circle Q is centered at C with radius 10 cm and also passes through B and D. Both quarter circles lie inside the square, and the region they share is shaded. Find the area of the shaded region, to the nearest tenth of a square centimeter.
- A.
  21.5
- B.
  42.9
- C.
  57.1
- D.
  78.5
Answer: C. 57.1
Each quarter circle has area 25π ≈ 78.5 and together they cover the whole square, so their overlap is 2(25π) − 100 = 50π − 100 ≈ 57.1. Choice A (21.5) is the square minus one quarter circle; B (42.9) is the unshaded part of the square; D (78.5) is a whole quarter circle.

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