Question 1
A right circular cone has a radius of 7 inches and a height of 9 inches. What is its volume, in cubic inches?
- A.
21π
- B.
441π
- C.
147π
- D.
294π
Answer: C. 147π
V = (1/3)πr²h = (1/3)π(7)²(9) = 147π.
SAT · Math · Geometry and Trigonometry
Area and volume
Area and volume is part of Geometry and Trigonometry in the SAT 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 composite or shaded region area, cylinder volume and liquid level, cone, sphere or pyramid volume and rectangular prism surface area and diagonals.
Common traps: adding the removed part; using diameter as radius; double-counting an overlap.
Sample questions
Question 2
A glass paperweight is shaped like a pyramid with a square base. Each edge of the base is 6 centimeters long, and the volume of the paperweight is 84 cubic centimeters. What is the height of the paperweight, in centimeters?
Enter your own answer.
Answer: 7
The base area is 6² = 36 square centimeters, so V = (1/3)(36)h = 12h. Setting 12h = 84 gives h = 7. Forgetting the one-third factor gives 84 ÷ 36 ≈ 2.33, and using the base edge 6 in place of the base area gives (1/3)(6)h = 84, or h = 42.
Question 3
A cylindrical storage tank has an inside radius of 2 metres. Water is pumped into the tank at a constant rate of 500 litres per minute. Which of the following is closest to the number of minutes needed to raise the water level in the tank by 1.5 metres? (1 cubic metre = 1,000 litres; use π ≈ 3.14)
- A.
12.6
- B.
18.8
- C.
37.7
- D.
150.7
Answer: C. 37.7
The added water forms a cylinder of radius 2 m and height 1.5 m: volume = 3.14 × 2² × 1.5 = 18.84 cubic metres = 18,840 litres. At 500 litres per minute, the time is 18,840 ÷ 500 ≈ 37.7 minutes. B reports the volume in cubic metres as if it were minutes; D uses the diameter 4 as the radius; A uses the cone formula (1/3)πr²h.
- A.
Question 4
A circular coaster has a radius of 6 centimeters. A square tile is set into the coaster so that all four corners of the square touch the circular edge of the coaster. What is the area, in square centimeters, of the part of the coaster that is not covered by the tile?
- A.
72 − 18π
- B.
144 − 36π
- C.
36π − 72
- D.
18π
Answer: C. 36π − 72
The square's diagonal is a diameter, 12 centimeters, so its area is 12²/2 = 72. The coaster's area is 36π, which leaves 36π − 72 (C). 144 − 36π uses 12 as the square's side, the square drawn around the coaster; 72 − 18π finds the part of the square outside a circle inscribed in it; and 18π subtracts that inner circle instead of the square.
- A.
Question 5
A brass bar is a rectangular prism whose cross-section is a square 4 centimeters on each side. The bar has a mass of 1,088 grams, and the density of brass is 8.5 grams per cubic centimeter. How long is the bar, in centimeters?
- A.
8
- B.
32
- C.
68
- D.
128
Answer: A. 8
Volume = mass ÷ density = 1,088 ÷ 8.5 = 128 cubic centimeters. The square cross-section has area 4² = 16, so the length is 128 ÷ 16 = 8 (A). 128 is the volume itself, 32 divides by 4 instead of 4², and 68 divides the mass by 16 without first using the density.
- A.
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