20 medium SAT Area and volume questions

Medium is where most scores are actually won and lost. These questions are not tricky for the sake of it, but every one of them has a wrong answer built to catch a specific shortcut.

Every question below is a real item from the SAT Climb bank, tagged medium by the same difficulty model the app uses to build your practice. Pick an answer before you open the explanation.

Math · Geometry and Trigonometry~2 per testMedium tier

None of these repeat the 25 hard Area and volume questions, so the two pages together are 45 different questions on this one skill.

Question 1Medium

A right circular cylinder has a radius of 5 inches and a height of 12 inches. What is the volume, in cubic inches, of the cylinder? (The volume of a right circular cylinder is $\pi r^{2}$ h, where r is the radius and h is the height.)

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Why B is right

The volume of a right circular cylinder is given by the formula $V = \pi r^{2}$ h. Substituting $r = 5$ inches and $h = 12$ inches yields $V = \pi (5^{2}) (12) = \pi$ (25) $(12) = 300\pi$ cubic inches.

Why the others are wrong

AThis is the result of using the diameter instead of the radius squared, calculating π(10)(12) instead of π(5²)(12).
CThis is the result of adding the lateral surface area formula 2πrh = 2π(5)(12) = 120π to the base area πr² = 25π incorrectly.
DThis is the result of multiplying the radius by the height twice, calculating π(5)(12)(2) or confusing the formula with π(2r)h(2).

Question 2Medium

A rectangular box has a length of 10 inches, a width of 8 inches, and a height of 5 inches. A second box has dimensions that are each double the dimensions of the first box. What is the volume, in cubic inches, of the second box?

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Why D is right

The first box has volume 10 × 8 × 5 = 400 cubic inches. When dimensions are doubled, the new dimensions are 20, 16, and 10 inches. The volume of the second box is 20 × 16 × 10 = 3200 cubic inches. Alternatively, doubling all linear dimensions scales volume by $2^{3} = 8$ , so 400 × 8 = 3200.

Why the others are wrong

AThis is the volume of the original box, not accounting for the doubling of dimensions.
BThis results from doubling the original volume (400 × 2 = 800) instead of recognizing that volume scales by the cube of the linear scale factor.
CThis results from quadrupling the original volume (400 × 4 = 1600), which would be correct if only two dimensions were doubled or if calculating area scaling.

Question 3Medium

A circle has an area of $49\pi$ square feet. What is the circumference, in feet, of the circle?

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Why B is right

The area of a circle is $\pi r^{2}$ , so $\pi r^{2}$ $= 49\pi$ . Dividing both sides by $\pi$ gives $r^{2}$ $= 49$ , so $r = 7$ feet. The circumference is $2\pi r = 2\pi (7) = 14\pi$ feet.

Why the others are wrong

AThis results from confusing the radius (7 feet) with the circumference formula, computing πr instead of 2πr.
CThis is the numerical value from the area (49) without the π, or computing 2r · π incorrectly.
DThis is the radius rather than the circumference.

Question 4Medium

A rectangular garden has a length of 18 feet and a width of 15 feet. A concrete path 2 feet wide is built around the entire perimeter of the garden. What is the area, in square feet, of the concrete path?

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Why C is right

The outer dimensions including the path are (18 + 2 + 2) = 22 feet by (15 + 2 + 2) = 19 feet. The outer area is 22(19) = 418 square feet. The inner garden area is 18(15) = 270 square feet. The path area is 418 - 270 = 148 square feet.

Why the others are wrong

AThis is the result of calculating the perimeter of the outer rectangle 2(22 + 19) = 132, confusing perimeter with area.
BThis is the result of only adding the path width to one side of each dimension, calculating 20(17) - 18(15) = 340 - 270 = 70, then doubling incorrectly, or using (18 + 15)(2)(2) = 132 then adding 14.
DThis is the original garden area 18(15) = 270, not the path area.

Question 5Medium

A cylinder has a radius of 5 inches and a height of 12 inches. What is the volume, in cubic inches, of the cylinder? (The volume of a cylinder is equal to $\pi r^{2}$ h, where r is the radius and h is the height.)

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Why B is right

The volume of a cylinder is $\pi r^{2}$ h. Substituting $r = 5$ inches and $h = 12$ inches gives $\pi (5^{2}) (12) = \pi (25) (12) = 300\pi$ cubic inches.

Why the others are wrong

AThis incorrectly uses πrh instead of πr²h, giving π(5)(12) = 60π.
CThis swaps the squaring, using π(r)(h²) = π(5)(144) = 720π, or confuses dimensions to reach 120π.
DThis adds dimensions inappropriately, such as 2πr + πrh or another perimeter-based formula confusion.

Question 6Medium

A right triangle has legs of length 15 centimeters and 20 centimeters. What is the area, in square centimeters, of the triangle?

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Why B is right

The area of a triangle is (1/2)bh, where b and h are the base and height. For a right triangle, the two legs serve as base and height. Therefore, $\displaystyle A = \frac{1}{2}$ (15)(20) = 150 square centimeters.

Why the others are wrong

AThis results from incorrectly adding the two legs and subtracting half: (15 + 20) × 2 = 70, or from a perimeter-based miscalculation.
CThis results from incorrectly averaging the two legs and multiplying by a factor: (15 + 20)/2 × 10 = 175, or from a calculation error.
DThis results from forgetting the 1/2 factor and calculating 15 × 20 = 300 directly.

Question 7Medium

A sphere has a radius of 9 inches. If the radius is tripled, by what factor does the volume of the sphere increase? (The volume of a sphere is equal to $\displaystyle \frac{4}{3}\pi r^{3}$ , where r is the radius of the sphere.)

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Why C is right

When the radius is scaled by a factor of $k = 3$ , the volume scales by $k^{3}$ = $3^{3} = 27$ . The original volume is $\displaystyle \frac{4}{3}\pi (9^{3})$ and the new volume is $\displaystyle \frac{4}{3}\pi (27^{3})$ $\displaystyle = \frac{4}{3}\pi (3 \times 9)^{3} = 27 \times \frac{4}{3} \pi (9^{3})$ .

Why the others are wrong

AThis represents the linear scaling factor rather than the volume scaling factor.
BThis is k² = 9, the factor by which area would scale, not volume.
DThis results from incorrectly calculating 2 × 9 or 3 × 6.

Question 8Medium

A circle has an area of $121\pi$ square feet. What is the circumference, in feet, of the circle?

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Why B is right

From the area formula $A = \pi r^{2}$ , setting $\pi r^{2}$ $= 121\pi$ gives $r^{2}$ $= 121$ , so $r = 11$ feet. The circumference is $C = 2\pi r = 2\pi (11) = 22\pi$ feet.

Why the others are wrong

AThis is the radius (11) multiplied by π, but circumference requires 2πr, not πr.
CThis incorrectly doubles the correct circumference, possibly from confusing radius and diameter in the calculation.
DThis represents the area value itself rather than computing the circumference from the radius.

Question 9Medium

A rectangular garden has a length of 18 feet and a width of 12 feet. A uniform border of width 2 feet surrounds the garden. What is the area, in square feet, of the border?

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Why C is right

The outer rectangle has dimensions 18 + 2(2) = 22 feet by 12 + 2(2) = 16 feet, with area 22 × 16 = 352 square feet. The garden area is 18 × 12 = 216 square feet. The border area is 352 - 216 = 136 square feet.

Why the others are wrong

AThis results from incorrectly calculating 2(2)(18 + 12) = 4(30) = 120, then making an arithmetic error, or from miscalculating the perimeter contribution.
BThis results from calculating 2 × 2 × (18 + 12) = 4 × 30 = 120, then adding an incorrect adjustment, or from halving the correct answer.
DThis results from calculating only the outer rectangle area 22 × 16 = 352 without subtracting the garden area.

Question 10Medium

A sphere has a volume of $288\pi$ cubic centimeters. What is the radius, in centimeters, of the sphere? (The volume of a sphere is equal to $\displaystyle \frac{4}{3}\pi r^{3}$ , where r is the radius.)

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Why A is right

The volume formula is $\displaystyle \frac{4}{3}\pi r^{3}$ $= 288\pi$ . Dividing both sides by $\pi$ gives $\displaystyle \frac{4}{3} r^{3}$ $= 288$ . Multiplying by $3/4$ gives $r^{3}$ $= 216$ , so $r = 6$ centimeters.

Why the others are wrong

BThis incorrectly takes the cube root of 288 directly instead of first dividing by 4/3, or confuses the calculation steps.
CThis doubles the correct radius, possibly from confusing radius with diameter.
DThis uses the intermediate calculation 288 ÷ 4 = 72 without completing the cube root operation.

Question 11Medium

A right triangle has legs of length 15 inches and 20 inches. What is the area, in square inches, of the triangle?

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Why C is right

The area of a triangle is (1/2)bh, where b and h are the base and height. For a right triangle, the two legs serve as base and height. Substituting $b = 15$ and $h = 20$ yields (1/2)(15)(20) = 150 square inches.

Why the others are wrong

AThis results from computing the sum or semi-perimeter 15 + 20 + 25 (hypotenuse) and dividing by 2.
BThis results from omitting the factor of 1/2, computing 15 · 20 = 300 directly.
DThis results from computing the hypotenuse using the Pythagorean theorem (25 inches) and confusing it with area.

Question 12Medium

A rectangular field has a length of 120 meters and a width of 80 meters. If both the length and width are increased by 50%, what is the area, in square meters, of the enlarged field?

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Why D is right

Increasing each dimension by 50% means multiplying by 1.5. The new length is 120 × 1.5 = 180 meters and the new width is 80 × 1.5 = 120 meters. The area is 180 × 120 = 21600 square meters.

Why the others are wrong

AThis results from incorrectly adding 50% to only the total area (9600 × 1.5 = 14400) rather than scaling both dimensions individually.
BThis results from adding 50% of the original perimeter to the original area or other computational errors involving perimeter-like calculations.
CThis results from doubling the original area (9600 × 2 = 19200) instead of applying the 1.5 factor to each dimension.

Question 13Medium

A square has a perimeter of 52 meters. What is the area, in square meters, of the square?

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Why A is right

The perimeter of a square is 4s, where s is the side length. So $4s = 52$ , giving $s = 13$ meters. The area is $s^{2}$ = $13^{2} = 169$ square meters.

Why the others are wrong

BThis is the side length (13 meters) rather than the area.
CThis results from dividing the perimeter by 2 instead of squaring the side length.
DThis results from computing 13 · 26 or doubling the area through error.

Question 14Medium

A circular pool has a diameter of 18 feet. A deck extends 3 feet beyond the edge of the pool on all sides. What is the area, in square feet, of the deck? (The area of a circle is $\pi r^{2}$ , where r is the radius.)

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Why B is right

The pool has diameter 18 feet, so radius 9 feet. The deck extends 3 feet beyond, making the outer radius 12 feet. The area of the deck is the outer circle minus the pool: $\pi (12)^{2} - \pi (9)^{2} = 144\pi - 81\pi = 63\pi$ square feet.

Why the others are wrong

AThis results from using the deck width times the pool radius (3 × 9 = 27), which incorrectly treats this as a perimeter problem.
CThis is the area of the pool itself (π(9)² = 81π), not the area of the deck.
DThis results from doubling the correct answer, possibly from counting the deck area twice.

Question 15Medium

A sphere has a volume of $288\pi$ cubic feet. What is the radius, in feet, of the sphere? (The volume of a sphere is $\displaystyle \frac{4}{3}\pi r^{3}$ , where r is the radius.)

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Why A is right

Setting $\displaystyle \frac{4}{3}\pi r^{3}$ = $288\pi$ and dividing both sides by $\pi$ gives $\displaystyle \frac{4}{3} r^{3}$ = 288. Multiplying both sides by $3/4$ yields $r^{3}$ = 216. Taking the cube root of both sides gives $r = 6$ feet.

Why the others are wrong

BThis results from incorrectly solving r³ = 288 directly instead of first simplifying to r³ = 216.
CThis results from dividing 288 by 24 instead of correctly isolating r³.
DThis results from dividing 288 by 16 or using surface area formula instead of volume.

Question 16Medium

A circle has a radius of 14 centimeters. What is the area, in square centimeters, of the circle?

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Why C is right

The area of a circle is equal to $\pi r^{2}$ , where r is the radius. Substituting 14 centimeters for r yields $\pi (14^{2})$ $= 196\pi$ square centimeters.

Why the others are wrong

AThis is the circumference formula result 2πr = 2π(14) = 28π, not the area.
BThis results from incorrectly calculating half the area (196π ÷ 2 = 98π).
DThis results from incorrectly doubling the area (196π × 2 = 392π).

Question 17Medium

A rectangular storage container has a length of 15 inches, a width of 10 inches, and a height of 6 inches. If each dimension is doubled, by what factor does the volume of the container increase?

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Why D is right

When all linear dimensions of a three-dimensional figure are scaled by a factor of k, the volume is scaled by $k^{3}$ . Since each dimension is doubled $(k = 2)$ , the volume increases by a factor of $2^{3} = 8$ .

Why the others are wrong

AThis is the scaling factor for the linear dimensions, not the volume.
BThis is the scaling factor for area when linear dimensions are doubled (2²), not volume.
CThis incorrectly adds the three dimensions together (2 + 2 + 2 = 6) instead of using the cubic relationship.

Question 18Medium

A cube has a volume of 1728 cubic centimeters. What is the surface area, in square centimeters, of the cube?

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Why D is right

The volume of a cube is $s^{3}$ , where s is the edge length. Since $s^{3}$ = 1728, $s = 12$ centimeters. The surface area of a cube is $6s^{2}$ , so surface area = $6(12^{2})$ = 6(144) = 864 square centimeters.

Why the others are wrong

AThis results from finding s² = 144 without multiplying by 6, calculating only one face area instead of all six.
BThis results from incorrectly calculating 2 × 12 × 12 = 288, treating it like a perimeter calculation or doubling one face.
CThis results from calculating 4s² = 4(144) = 576 instead of 6s², missing two faces of the cube.

Question 19Medium

A rectangular garden has length 18 feet and width 14 feet. A walkway 2 feet wide surrounds the garden. What is the area, in square feet, of the walkway?

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Why C is right

The outer rectangle including the walkway has dimensions 22 feet by 18 feet (adding 2 feet on each side). Its area is 396 square feet. The garden area is 18 × 14 = 252 square feet. The walkway area is 396 - 252 = 144 square feet.

Why the others are wrong

AThis is the perimeter of the walkway boundaries (2 × 32), not its area.
BThis incorrectly calculates only the area added on two sides of the garden instead of all four sides.
DThis is the total area of the outer rectangle including both garden and walkway, not just the walkway.

Question 20Medium

A rectangular garden has a length of 20 feet and a width of 12 feet. If both dimensions are doubled, what is the area, in square feet, of the new garden?

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Why B is right

When both dimensions are doubled, the new length is 40 feet and the new width is 24 feet. The area is 40 × 24 = 960 square feet. Alternatively, when linear dimensions are scaled by $k = 2$ , area scales by $k^{2}$ = 4, so the new area is 4 × 240 = 960 square feet.

Why the others are wrong

AThis results from doubling the original area (240 × 2 = 480) instead of quadrupling it.
CThis results from doubling the perimeter (64 × 2 = 128) instead of calculating area.
DThis results from incorrectly scaling by a factor of 2.67 or miscalculating 40 × 24.

What to do after medium

Medium is the tier that decides most scores. If these are landing, the hard set is where the remaining points are.

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Questions are written by SAT Climb and drawn from its own item bank. SAT® is a registered trademark of College Board, which is not affiliated with and does not endorse SAT Climb.

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