Area, Perimeter, and Volume - Quantitative Reasoning

A cone has radius 3 cm and height 12 cm. What is the volume of the cone, in cubic centimeters?
Answer: $36\pi$ 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.
A rectangle has length $12$ cm and width $5$ cm. What is the area of the rectangle, in square centimeters?
Answer: 60 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).
A square has perimeter $36$ cm. What is the area of the square, in square centimeters?
Answer: 81 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).
A circle's radius is increased by 50%, from $r$ to $1.5r$ . By what factor does the area of the circle increase?
Answer: 2.25 This question tests how area scales with radius increase. Area is πr²; increasing r by 50% to 1.5r gives new area π(1.5r)² = 2.25πr², factor 2.25. This squares the scaling factor 1.5. The result is justified by the area formula. A distractor like 1.5 might use linear scaling for area. Another like 3.0 could double incorrectly.
A square has area $81$ square units. What is the perimeter of the square, in units?
Answer: $36$ 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.
A square has side length $s$ . A second square has side length $2s$ . What is the ratio of the area of the second square to the area of the first square?
Answer: 4 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.
A circle has circumference $18\pi$ cm. What is the area of the circle, in square centimeters?
Answer: $81\pi$ 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.
A square poster has side length $20$ cm. If each side length is decreased by $25\%$ , what is the area of the resulting square, in square centimeters?
Answer: $225$ 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.
A right triangle has legs of lengths $6$ m and $8$ m. What is the area of the triangle, in square meters?
Answer: 24 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).
A rectangle has length $9$ ft and width $4$ ft. If both the length and width are doubled, what is the area of the new rectangle, in square feet?
Answer: 144 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.
A cylindrical tank has radius $3$ m and height $10$ m. What is the volume of the tank, in cubic meters? (Express your answer in terms of $\pi$ .)
Answer: $90\pi$ 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.
A cube has volume 125 cubic inches. What is the total surface area of the cube, in square inches?
Answer: 150 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.
A right circular cylinder has radius $3$ inches and height $10$ inches. What is the volume of the cylinder, in cubic inches?
Answer: $90\pi$ 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.
A circle has radius $7$ cm. What is the area of the circle, in square centimeters?
Answer: $49\pi$ 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.
A square has side length $s$ . If the side length is increased by $50\%$ , what is the percent increase in the area of the square?
Answer: 125% 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.
A semicircle has diameter $10$ meters. What is the perimeter of the semicircle, in meters? (Perimeter includes the diameter.)
Answer: $10+5\pi$ 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).
A square and a rectangle have the same perimeter. The square has side length 7 cm. The rectangle has length 10 cm. What is the width of the rectangle, in centimeters?
Answer: 4 This question tests perimeter relationships between shapes. Perimeter of square is 4×side; for rectangle, 2(length + width). Square side 7 cm gives perimeter 28 cm, so rectangle: 2(10 + w) = 28, 10 + w = 14, w = 4 cm. This equates perimeters and solves for width. The result is justified by the shared perimeter. A distractor like 7 might confuse side lengths. Another like 14 could be half the perimeter.
A cube has side length $5$ inches. What is the total surface area of the cube, in square inches?
Answer: 150 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).
A square has perimeter 48 inches. What is the area of the square, in square inches?
Answer: 144 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.
A circle has circumference $30\pi$ centimeters. What is the area of the circle, in square centimeters?
Answer: $225\pi$ 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.

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