SAT Math Quiz: Area And Volume

What this quiz covers

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

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. $8\text{ units}^2$
2. $10\text{ units}^2$
3. $12\text{ units}^2$ (correct answer)
4. $24\text{ units}^2$

Explanation: This problem asks for the area of a triangle given coordinates of its vertices in square units. Since the triangle has a horizontal base from A(1,1) to B(7,1), the base length is 7 - 1 = 6 units. The height is the vertical distance from this base to C(1,5), which is 5 - 1 = 4 units. Using the formula Area = ½ × base × height: Area = ½ × 6 × 4 = 12 square units. A common error is forgetting to multiply by ½, which would give 24 square units. When working with coordinate geometry, sketch the triangle to visualize the base and height.

Question 2

1. $98\text{ in}^2$
2. $182\text{ in}^2$
3. $91\text{ in}^2$ (correct answer)
4. $364\text{ in}^2$

Explanation: This problem asks for the area of a trapezoid in square inches. The formula for area of a trapezoid is A = ½ × (base₁ + base₂) × height. Substituting the given values: A = ½ × (10 + 16) × 7 = ½ × 26 × 7 = 91 in². A common error is forgetting to divide by 2, which would give 182 in². When working with trapezoids, remember to add both bases before multiplying by height and dividing by 2.

Question 3

1. $120\text{ in}^3$
2. $240\text{ in}^3$
3. $400\text{ in}^3$ (correct answer)
4. $800\text{ in}^3$

Explanation: This problem asks for the volume of a square pyramid in cubic inches. The formula for volume of a pyramid is V = (1/3) × base area × height. The base is a square with side 10 in, so base area = 10² = 100 in². Volume = (1/3) × 100 × 12 = (1/3) × 1200 = 400 in³. A common error is forgetting to divide by 3, which would give 1200 in³. Remember that pyramid volume is always one-third of the prism volume with the same base and height.

Question 4

1. $140\text{ ft}^2$
2. $45.5\text{ ft}^2$
3. $182\text{ ft}^2$
4. $91\text{ ft}^2$ (correct answer)

Explanation: This problem asks for the area of a parallelogram in square feet. The formula for area of a parallelogram is A = base × height. Substituting the given values: A = 13 × 7 = 91 ft². A common error is using the slant side length instead of the perpendicular height, or trying to use a triangle formula. Remember that for parallelograms, you need the perpendicular height, not the side length.

Question 5

1. $37.7\text{ cm}^3$
2. $452.4\text{ cm}^3$
3. $113.0\text{ cm}^3$
4. $150.7\text{ cm}^3$ (correct answer)

Explanation: The question asks for the volume of a conical paper cup with radius 4 cm and height 9 cm, rounded to the nearest tenth of a cubic centimeter. The formula needed is V = (1/3) π r² h, selecting it for cones and noting cm³ units. Substitute r=4 cm, h=9 cm, π≈3.14: r²=16, π r²=3.14×16=50.24, then 50.24×9=452.16, and (1/3)×452.16=150.72, rounding to 150.7 cm³. Perform multiplications step by step for accuracy. A key error is using the full cylinder formula without the 1/3 factor. Another mistake could be confusing radius and height. A strategy is to recall that cone volume is one-third of a cylinder with same base and height.

Question 6

1. $2.12\text{ m}^3$
2. $1.77\text{ m}^3$
3. $13.57\text{ m}^3$
4. $4.24\text{ m}^3$ (correct answer)

Explanation: The question asks for the volume of a cylindrical water tank in cubic meters, rounded to the nearest hundredth. The formula needed is V = π r² h, with careful attention to units in meters. Substitute r = 0.75 m, h = 2.4 m, and π ≈ 3.14 into the formula: first, r² = (0.75)² = 0.5625. Then, π r² = 3.14 × 0.5625 = 1.76625, and V = 1.76625 × 2.4 = 4.239, which rounds to 4.24 m³. A key error might be forgetting to square the radius, resulting in an incorrect volume. Another mistake could be using the diameter instead of the radius. A test-taking strategy is to perform calculations step by step and verify rounding only at the end for accuracy.

Question 7

1. 120π cm³
2. 180π cm³
3. 360π cm³
4. 90π cm³ (correct answer)

Explanation: This question asks for the volume of a cylinder with height 10 cm and diameter 6 cm. The formula for cylinder volume is V = πr²h. Since the diameter is 6 cm, the radius is 3 cm. Substituting: V = π(3)²(10) = π(9)(10) = 90π cm³.

Question 8

1. 63 cm$^2$
2. 85.5 cm$^2$ (correct answer)
3. 108 cm$^2$
4. 153 cm$^2$

Explanation: This question asks for the area of a composite figure: a rectangle with a triangle removed. The rectangle has area 12 × 9 = 108 square centimeters. The removed right triangle has legs 5 cm and 9 cm, so its area is ½(5)(9) = 22.5 square centimeters. The remaining area is 108 - 22.5 = 85.5 square centimeters. A key insight is recognizing that the triangle shares the rectangle's sides, making calculation straightforward. For composite figures with removed sections, calculate the whole then subtract.

Question 9

1. 12 yd$^2$ (correct answer)
2. 36 yd$^2$
3. 108 yd$^2$
4. 324 yd$^2$

Explanation: This question asks for the area of a rectangular floor in square yards, given dimensions in feet. First, find the area in square feet: Area = 12 ft × 9 ft = 108 square feet. Since 1 yard = 3 feet, we have 1 square yard = 9 square feet. Converting: 108 ÷ 9 = 12 square yards. A critical error is converting linear feet to yards (dividing by 3) instead of square feet to square yards (dividing by 9). Always square the conversion factor when converting area units.

Question 10

1. 20 cm$^2$
2. 45.5 cm$^2$
3. 91 cm$^2$ (correct answer)
4. 182 cm$^2$

Explanation: This question asks for the area of a parallelogram with base 13 cm and height 7 cm. The area of a parallelogram is A = base × height. Substituting: A = 13 × 7 = 91 square centimeters. A common mistake is confusing the height with a slanted side length - the height must be perpendicular to the base. Unlike triangles, parallelograms don't have the ½ factor in their area formula.

Question 11

1. $0.75\text{ ft}^3$
2. $1.25\text{ ft}^3$ (correct answer)
3. $2.50\text{ ft}^3$
4. $15\text{ ft}^3$

Explanation: This problem asks for the volume of a rectangular prism in cubic feet, given dimensions in inches. The formula for volume of a rectangular prism is $V = \text{length} \times \text{width} \times \text{height}$. First, we must convert all dimensions from inches to feet: $18 \text{ in} = \frac{18}{12} = 1.5 \text{ ft}$, $12 \text{ in} = \frac{12}{12} = 1 \text{ ft}$, and $10 \text{ in} = \frac{10}{12} = \frac{5}{6} \text{ ft}$. Substituting into the formula: $V = 1.5 \times 1 \times \frac{5}{6} = 1.5 \times \frac{5}{6} = \frac{7.5}{6} = 1.25 \text{ ft}^3$. A common error is forgetting to convert all three dimensions or calculating in cubic inches then dividing by 12 instead of $12^3$. When dealing with unit conversions for volume, remember to convert each linear dimension separately before multiplying.

Question 12

1. $55\text{ ft}^2$ (correct answer)
2. $110\text{ ft}^2$
3. $22\text{ ft}^2$
4. $70\text{ ft}^2$

Explanation: This problem asks for the area of a trapezoid with bases 14 ft and 8 ft and height 5 ft. The formula for area of a trapezoid is A = (1/2) × (b₁ + b₂) × h. Substituting: A = (1/2) × (14 + 8) × 5 = (1/2) × 22 × 5 = (1/2) × 110 = 55 ft². A common error is forgetting to divide by 2, giving 110 ft², or only using one base. Remember that trapezoid area uses the average of the two bases multiplied by height.

Question 13

1. $900\text{ cm}^3$
2. $360\text{ cm}^3$ (correct answer)
3. $720\text{ cm}^3$
4. $180\text{ cm}^3$

Explanation: This problem asks for the volume of a triangular prism with right triangle base (legs 6 cm and 8 cm) and length 15 cm. The volume formula is V = Base Area × Length. Base area = (1/2) × 6 × 8 = 24 cm². Volume = 24 × 15 = 360 cm³. A common error is forgetting to find the base area first or using the wrong formula for triangular area. For prisms, always find the base area first, then multiply by the prism's length or height.

Question 14

1. $80+8\pi\text{ ft}^2$ (correct answer)
2. $80+16\pi\text{ ft}^2$
3. $40+16\pi\text{ ft}^2$
4. $88\pi\text{ ft}^2$

Explanation: This problem asks for the area of a composite shape: a 10 ft × 8 ft rectangle with a semicircle attached along the 8 ft side. The area equals the rectangle area plus the semicircle area. Rectangle area = 10 × 8 = 80 ft². For the semicircle, diameter = 8 ft, so radius = 4 ft, and semicircle area = ½πr² = ½π(4)² = ½π(16) = 8π ft². Total area = 80 + 8π ft². A common error is calculating the full circle area instead of half, which would give 80 + 16π ft².

Question 15

1. $188.4\text{ cm}^3$
2. $251.2\text{ cm}^3$ (correct answer)
3. $502.4\text{ cm}^3$
4. $753.6\text{ cm}^3$

Explanation: The question asks for the volume of a cone with height 15 cm and radius 4 cm, in cubic centimeters, using π ≈ 3.14 and rounding to the nearest tenth if needed. The formula for cone volume is V = (1/3) π r² h. Substitute: r² = 16, then π × 16 = 3.14 × 16 = 50.24, times h=15 gives 753.6, then divide by 3: 753.6 / 3 = 251.2 cm³. A frequent mistake is forgetting the 1/3 factor, yielding π r² h = 753.6 cm³ instead. Emphasize including all parts of the formula for accuracy. Units in centimeters result in cubic centimeters. A strategy is to write the formula first and plug in values one at a time.

Question 16

1. $74.1\text{ m}^2$ (correct answer)
2. $88.3\text{ m}^2$
3. $60.0\text{ m}^2$
4. $102.5\text{ m}^2$

Explanation: The question asks for the total area of a garden shaped as a rectangle 10 m by 6 m with a semicircle of diameter 6 m attached to one shorter side, in square meters, using π ≈ 3.14. The total area is the rectangle area plus the semicircle area; rectangle area = length × width, semicircle = (1/2) π r² where r = diameter/2 = 3 m. Calculate rectangle: 10 × 6 = 60 m²; semicircle: (1/2) × 3.14 × 9 = 14.13 m²; total = 60 + 14.13 = 74.13 ≈ 74.1 m². Ensure the semicircle's diameter matches the attached side (6 m) to avoid miscalculating the radius. Units are consistent in meters, yielding square meters. A test-taking tip is to break composite shapes into familiar parts and add their areas carefully.

Question 17

1. $132\text{ in}^2$
2. $66\text{ in}^2$ (correct answer)
3. $44\text{ in}^2$
4. $72\text{ in}^2$

Explanation: The question asks for the area of a trapezoid with bases 14 in and 8 in, and height 6 in, in square inches. The formula for trapezoid area is (1/2) × (b1 + b2) × h, where b1 and b2 are the bases and h is the height. Substitute: (1/2) × (14 + 8) × 6 = (1/2) × 22 × 6 = 11 × 6 = 66 in². A common error is averaging the bases incorrectly, like (14 + 8)/2 = 11 but then forgetting to multiply by height, or multiplying all three numbers 14×8×6=672. Always apply the formula step-by-step to avoid missteps. Units in inches give area in square inches. For tests, recall that the height must be perpendicular to the bases.

Question 18

1. $12.5\text{ units}^2$
2. $30\text{ units}^2$
3. $25\text{ units}^2$
4. $15\text{ units}^2$ (correct answer)

Explanation: The question asks for the area of triangle ABC with vertices A(1,1), B(7,1), and C(1,6), in square units. The formula for the area of a triangle is (1/2) × base × height, where base and height are perpendicular. Here, the base AB is from (1,1) to (7,1), so length 6 units along the x-axis; the height is the perpendicular distance from C to this base, which is the vertical difference from y=1 to y=6, or 5 units. Thus, area = (1/2) × 6 × 5 = 15 square units. A common mistake is using the slanted side AC or BC as the height instead of the perpendicular, which would give an incorrect area; always ensure the height is perpendicular to the base. Units are consistent as coordinate units, resulting in square units. For test-taking, plot the points quickly to visualize the base and height accurately.

Question 19

1. 94.2 ft$^3$
2. 282.6 ft$^3$ (correct answer)
3. 314.0 ft$^3$
4. 942.0 ft$^3$

Explanation: This question asks for the volume of a cylindrical water tank in cubic feet. The volume of a cylinder is given by V = πr²h, where r is the radius and h is the height. Substituting the given values: V = π(3²)(10) = π(9)(10) = 90π cubic feet. Using π ≈ 3.14, we get V = 90(3.14) = 282.6 cubic feet. A common error is using diameter instead of radius, which would give 1130.4 cubic feet. When dealing with cylinders, always verify whether you're given radius or diameter.

Question 20

1. $40\pi$
2. $54\pi$
3. $72\pi$
4. $80\pi$ (correct answer)

Explanation: volume = $\pi r^{2}h$ = $\pi \cdot 4^{2} \cdot 5$ = $80\pi$