GRE Quiz: Area Perimeter Volume

What this quiz covers

This quiz focuses on Area Perimeter Volume, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $9\pi$
2. $18\pi$
3. $36\pi$
4. $72\pi$
5. $81\pi$ (correct answer)

Explanation: This question tests finding area given circumference of a circle. The circumference formula is C = 2πr, so with C = 18π, we have 2πr = 18π, giving r = 9 cm. The area formula is A = πr², so A = π(9)² = 81π square centimeters. This matches answer choice E. A common error would be using the radius value directly without squaring it, giving A = 9π (choice A), or confusing radius with diameter calculations.

Question 2

1. 50%
2. 75%
3. 100%
4. 125% (correct answer)
5. 150%

Explanation: This question tests percent change in area when dimensions change. For a square with side length s, the area is A₁ = s². When the side length increases by 50%, the new side length is 1.5s, and the new area is A₂ = (1.5s)² = 2.25s². The percent increase in area is [(2.25s² - s²)/s²] × 100% = [1.25s²/s²] × 100% = 125%. This matches answer choice D. A common error is assuming that a 50% increase in side length yields a 50% increase in area (choice A), failing to account for the quadratic relationship between linear dimensions and area.

Question 3

1. $14\pi$
2. $28\pi$
3. $49\pi$ (correct answer)
4. $98\pi$
5. $196\pi$

Explanation: This question tests the area formula for a circle. The area of a circle with radius r is A = πr², so with radius 7 cm, the area is A = π(7)² = 49π square centimeters. This formula represents the fundamental relationship between a circle's radius and its enclosed area. The area grows quadratically with the radius, not linearly. A common mistake is to use the circumference formula 2πr instead, which would give 14π, or to forget to square the radius and calculate 7π.

Question 4

1. $30\pi$
2. $60\pi$
3. $90\pi$ (correct answer)
4. $180\pi$
5. $300\pi$

Explanation: This question tests the volume formula for a right circular cylinder. The volume of a cylinder is V = πr²h, where r is the radius and h is the height. With radius 3 inches and height 10 inches, the volume is V = π(3)²(10) = π(9)(10) = 90π cubic inches. This formula represents the area of the circular base (πr²) multiplied by the height. A common error is to use the diameter instead of the radius, which would give π(6)²(10) = 360π, or to confuse the formula with surface area calculations.

Question 5

1. $9$
2. $18$
3. $27$
4. $36$ (correct answer)
5. $72$

Explanation: This question tests the relationship between area and perimeter of a square. For a square with area A, each side has length s = √A, so with area 81, each side is √81 = 9 units. The perimeter of a square is P = 4s = 4(9) = 36 units. The key insight is that area equals side squared, so we must take the square root to find the side length before calculating perimeter. A common error is to confuse area and perimeter formulas, perhaps dividing 81 by 4 to get 20.25 or multiplying 81 by 4 to get 324.

Question 6

1. $150$
2. $200$
3. $225$ (correct answer)
4. $300$
5. $400$

Explanation: This question tests how area scales when dimensions change by a percentage. If each side of the square is decreased by 25%, the new side length is 20 × (1 - 0.25) = 20 × 0.75 = 15 cm. The area of the new square is 15² = 225 square centimeters. Alternatively, since area scales as the square of the linear scaling factor, the new area is 400 × (0.75)² = 400 × 0.5625 = 225. A common error is to decrease the area by 25% directly, calculating 400 × 0.75 = 300, which incorrectly applies the percentage decrease to the area rather than to the linear dimensions.

Question 7

1. $36\pi$ (correct answer)
2. $108\pi$
3. $432\pi$
4. 108
5. $144\pi$

Explanation: This question tests the volume of a cone. The volume is (1/3)πr²h. Radius 3 cm, height 12 cm gives (1/3)π×9×12 = (1/3)π×108 = 36π cubic centimeters. This applies the formula directly. The result is justified by the cone's volume principle. A distractor like 108π might omit the 1/3 factor. Another like 432π could multiply extra.

Question 8

1. 144 (correct answer)
2. 192
3. 96
4. 12
5. 576

Explanation: This question tests the area of a square given its perimeter. The area of a square is side², and the side length is perimeter divided by 4. With a perimeter of 48 inches, the side is 48 / 4 = 12 inches, so the area is 12 × 12 = 144 square inches. This applies the relationship between perimeter and side directly. The result is justified as it follows from the square's properties. A distractor like 576 might result from squaring half the perimeter (24² = 576) due to a scaling misconception. Thus, the area is correctly 144.

Question 9

1. $225\pi$ (correct answer)
2. $900\pi$
3. $60\pi$
4. $15\pi$
5. 225

Explanation: This question tests the area of a circle given its circumference. The circumference is C = 2πr, so r = C / (2π); area is πr². Given C = 30π cm, r = 30π / (2π) = 15 cm, so area = π×15² = 225π square centimeters. This derives radius first then applies the area formula. The result is justified by the relationship between circumference and radius. A distractor like 900π might come from squaring 30 instead of 15. Another like 60π could be from using diameter incorrectly.

Question 10

1. 72
2. 144 (correct answer)
3. 26
4. 288
5. 104

Explanation: This question tests how area scales when dimensions change. The original rectangle has area = 9 × 4 = 36 square feet. When both length and width are doubled, the new dimensions are 18 ft × 8 ft, giving new area = 18 × 8 = 144 square feet. The area increases by a factor of 4 (2² = 4) when both dimensions double, confirming 36 × 4 = 144, which matches choice B. A common error would be thinking area only doubles (36 × 2 = 72, choice A) when dimensions double.

Question 11

1. 81 (correct answer)
2. 144
3. 9
4. 72
5. 324

Explanation: This question tests the relationship between perimeter and area of a square. For a square with perimeter P, each side has length s = P/4. With perimeter = 36 cm, each side is 36/4 = 9 cm. The area of a square is A = s², so A = 9² = 81 square centimeters. The result is 81 cm², which matches choice A. A common mistake would be confusing side length with area (using 9 as the answer, choice C) or incorrectly calculating 36 × 2 = 72 (choice D).

Question 12

1. $30\pi$
2. $60\pi$
3. $90\pi$ (correct answer)
4. $300\pi$
5. $900\pi$

Explanation: This question tests the volume of a cylinder. The volume formula for a cylinder is V = πr²h, where r is the radius and h is the height. With radius r = 3 cm and height h = 10 cm, we calculate V = π(3)²(10) = π(9)(10) = 90π cubic centimeters. This matches answer choice C. A common error would be using diameter instead of radius, giving V = π(6)²(10) = 360π, or forgetting to square the radius, yielding V = π(3)(10) = 30π (choice A).

Question 13

1. 25
2. 150 (correct answer)
3. 125
4. 100
5. 30

Explanation: This question tests the surface area of a cube. A cube has 6 identical square faces, and the surface area formula is SA = 6s², where s is the side length. With side length = 5 inches, we calculate: SA = 6 × 5² = 6 × 25 = 150 square inches. The result is 150 in², which matches choice B. A common mistake would be calculating the area of just one face (5² = 25, choice A) or finding the volume instead (5³ = 125, choice C).

Question 14

1. $90\pi$ (correct answer)
2. $60\pi$
3. $30\pi$
4. $13\pi$
5. $180\pi$

Explanation: This question tests the volume of a cylinder. The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius and h is the height. With radius = 3 m and height = 10 m, we calculate: V = π × 3² × 10 = π × 9 × 10 = 90π cubic meters. The result is 90π m³, which matches choice A. A common mistake would be using the circumference formula (2πrh = 2π × 3 × 10 = 60π, choice B) instead of the volume formula.

Question 15

1. $5\pi$
2. $10\pi$
3. $10+5\pi$ (correct answer)
4. $10+10\pi$
5. $20+5\pi$

Explanation: This question tests the perimeter of a semicircle including its diameter. A semicircle with diameter 10 meters has radius 5 meters. The curved part of the perimeter is half the circumference of a full circle: ½(2πr) = πr = 5π meters. The total perimeter includes this curved part plus the diameter: 5π + 10 meters. This can be written as 10 + 5π meters, matching answer choice C. A common error would be forgetting to include the diameter, giving only 5π (choice A).

Question 16

1. 34
2. 60 (correct answer)
3. 17
4. 120
5. 30

Explanation: This question tests the area of a rectangle. The area of a rectangle is calculated using the formula A = length × width. With length = 12 cm and width = 5 cm, we multiply: A = 12 × 5 = 60 square centimeters. The result is 60 cm², which matches choice B. A common error would be adding the dimensions (12 + 5 = 17, choice C) or finding the perimeter instead of area (2(12 + 5) = 34, choice A).

Question 17

1. 34
2. 120 (correct answer)
3. 17
4. 240
5. 40

Explanation: This question tests the volume of a rectangular prism. The volume of a rectangular prism is calculated using the formula V = length × width × height. With dimensions 10 cm × 4 cm × 3 cm, we multiply: V = 10 × 4 × 3 = 120 cubic centimeters. The result is 120 cm³, which matches choice B. A common error would be adding the dimensions (10 + 4 + 3 = 17, choice C) or calculating surface area instead of volume (2(10×4 + 10×3 + 4×3) = 164, not listed).

Question 18

1. 14
2. 48
3. 24 (correct answer)
4. 28
5. 96

Explanation: This question tests the area of a right triangle. The area of a triangle is calculated using the formula A = ½ × base × height, and for a right triangle, the legs serve as base and height. With legs of 6 m and 8 m, we calculate: A = ½ × 6 × 8 = ½ × 48 = 24 square meters. The result is 24 m², which matches choice C. A common error would be forgetting to divide by 2 (6 × 8 = 48, choice B) or adding the legs instead of multiplying (6 + 8 = 14, choice A).

Question 19

1. 150 (correct answer)
2. 25
3. 750
4. 375
5. 30

Explanation: This question tests surface area of a cube from its volume. Volume is side³, so side = cube root of volume; surface area is 6×side². Volume 125 gives side 5 inches, area 6×25 = 150 square inches. This derives side then applies formula. The result is justified by the cube's properties. A distractor like 750 might multiply volume by 6. Another like 25 could be side² only.

Question 20

1. 2
2. 4 (correct answer)
3. 8
4. $\tfrac{1}{2}$
5. $\tfrac{1}{4}$

Explanation: This question tests area ratio for scaled squares. Area is side², so ratio is (new side / old side)². Sides s and 2s give ratio (2s)² / s² = 4s² / s² = 4. This applies the scaling principle. The result is justified as areas scale with square of linear dimensions. A distractor like 2 might use linear scaling for area. Another like 8 could cube mistakenly.