Praxis Math Quiz: Compute Area And Volume

20 questions · exam conditions

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Compute Area And VolumeQuestion 1 of 20

In the figure, a circle is inscribed in a square. If the square has side length 10 inches, what is the area of the shaded region (inside the square but outside the circle), in square inches?

$Question graphic$

14.25

21.50

21.46

78.50

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Praxis Math Quiz

Praxis Math Quiz: Compute Area And Volume

Practice Compute Area And Volume in Praxis Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

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

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Question 1

In the figure, a circle is inscribed in a square. If the square has side length 10 inches, what is the area of the shaded region (inside the square but outside the circle), in square inches?

$14.25$
$21.50$ (correct answer)
$21.46$
$78.50$

Explanation: Square area =

10^2 = 100

in². Inscribed circle has radius 5 in, so circle area =

\pi(5)^2 = 25\pi \approx 25(3.14) = 78.50

in². Shaded area =

100 - 78.50 = 21.50

in². (A) uses incorrect radius or calculation. (C) uses a slightly different π approximation. (D) gives the circle area instead of the shaded area.

Question 2

A sphere is inscribed inside a cube as shown. If the cube has edge length 10 cm, what is the volume of the empty space inside the cube but outside the sphere, in cubic centimeters?

$476.67$ (correct answer)
$523.33$
$606.50$
$720.00$

Explanation: Cube volume =

10^3 = 1000

cm³. Sphere radius = 5 cm (half the edge). Sphere volume =

\tfrac{4}{3}\pi r^3 = \tfrac{4}{3}(3.14)(125) \approx 523.33

cm³. Empty space =

1000 - 523.33 \approx 476.67

cm³. (B) gives the sphere volume itself. (C) uses radius 10 (diameter as radius) in a combined miscalculation. (D) uses

\pi r^2

instead of

\tfrac{4}{3}\pi r^3

Question 3

A cylindrical water tank has a radius of 5 meters and is filled with water to a height of 8 meters. If the radius of the tank is doubled, but the volume of water remains the same, what will be the new height of the water in the tank?

2 meters (correct answer)
4 meters
1 meter
1.5 meters

Explanation: First, calculate the original volume of water: V = πr²h = π(5)²(8) = 200π cubic meters. The new radius is double the original, so the new radius is 10 meters. The volume of water is unchanged. Let the new height be h_new. The volume formula with the new radius is V = π(10)²(h_new) = 100π(h_new). Set the volumes equal: 200π = 100π(h_new). Divide both sides by 100π to solve for h_new, which gives h_new = 2 meters.

Question 4

A pizza with a 16-inch diameter is cut into 8 equal slices. A smaller pizza with a 12-inch diameter is cut into 6 equal slices. What is the positive difference in area between one slice of the larger pizza and one slice of the smaller pizza?

2π sq in (correct answer)
4π sq in
8π sq in
28π sq in

Explanation: For the larger pizza, the radius is 16/2 = 8 inches. Its total area is A = π(8)² = 64π sq in. The area of one slice is 64π / 8 = 8π sq in. For the smaller pizza, the radius is 12/2 = 6 inches. Its total area is A = π(6)² = 36π sq in. The area of one slice is 36π / 6 = 6π sq in. The positive difference between the slice areas is 8π - 6π = 2π sq in.

Question 5

A concrete pipe is a cylinder with a length of 10 feet. It has an outer diameter of 3 feet and an inner diameter of 2 feet. What is the volume of the concrete used to make the pipe, in cubic feet? (Use π ≈ 3.14)

15.7 cu ft
39.25 cu ft (correct answer)
70.65 cu ft
109.9 cu ft

Explanation: This problem requires finding the volume of a hollow cylinder by subtracting the inner volume from the outer volume. The outer radius is 3÷2 = 1.5 ft and the inner radius is 2÷2 = 1 ft. The length is 10 ft. The volume of the outer cylinder is V_outer = πr²h = 3.14 × (1.5)² × 10 = 3.14 × 2.25 × 10 = 70.65 cu ft. The volume of the inner hollow space is V_inner = πr²h = 3.14 × (1)² × 10 = 31.4 cu ft. The volume of concrete is the difference: 70.65 - 31.4 = 39.25 cubic feet.

Question 6

A rectangular garden measures 12 feet by 15 feet. Inside the garden, a circular fountain with a diameter of 6 feet is installed. What is the area of the garden remaining for planting, in square feet? (Use π ≈ 3.14)

151.74 sq ft (correct answer)
161.16 sq ft
174.00 sq ft
68.84 sq ft

Explanation: First, calculate the total area of the rectangular garden: Area = length × width = 15 ft × 12 ft = 180 sq ft. Next, calculate the area of the circular fountain. The diameter is 6 feet, so the radius is half of that, which is 3 feet. The area of a circle is A = πr². Using r = 3, the area is A = 3.14 × (3)² = 3.14 × 9 = 28.26 sq ft. To find the remaining area for planting, subtract the fountain's area from the garden's area: 180 sq ft - 28.26 sq ft = 151.74 sq ft.

Question 7

A block of wood is a rectangular prism with dimensions 2 inches by 4 inches by 5 inches. If a hole with a radius of 1 inch is drilled all the way through the block, parallel to the 5-inch side, what is the volume of the remaining wood? (Use π ≈ 3.14)

24.30 cu in (correct answer)
32.44 cu in
40.00 cu in
20.60 cu in

Explanation: First, calculate the initial volume of the rectangular block: V_block = 2 in × 4 in × 5 in = 40 cubic inches. Next, calculate the volume of the cylindrical hole that was removed. The hole has a radius of 1 inch and its height is the length of the side it passes through, which is 5 inches. The volume of the cylinder is V_hole = πr²h = 3.14 × (1)² × 5 = 3.14 × 5 = 15.7 cubic inches. The volume of the remaining wood is the initial volume minus the volume of the hole: 40 cu in - 15.7 cu in = 24.3 cubic inches.

Question 8

A right square pyramid is shown. Based on the figure, what is the total surface area of the pyramid, in square inches?

$144$
$180$
$240$ (correct answer)
$260$

Explanation: Base area =

12^2 = 144

in². Slant height given is 8 in. Each triangular face has area

\tfrac{1}{2}(12)(8) = 48

in². Four faces =

192

in². Wait: total =

144 + 192 = 336

. That doesn't match either. Recompute with base 8, slant 11: base = 64, lateral = 4·(½·8·11) = 176, total = 240. (A) base only plus half lateral. (B) uses slant height 8 with base 8: 64+128=192... (D) uses height instead of slant height incorrectly.

Question 9

A storage container is in the shape of the composite solid shown: a rectangular prism with a triangular prism (like a roof) on top. Based on the figure, what is the total volume of the container, in cubic feet?

$360$
$420$ (correct answer)
$480$
$540$

Explanation: Rectangular prism volume =

10 \times 6 \times 5 = 300

ft³. Triangular prism (roof) volume = (area of triangular cross-section) × length =

\frac{1}{2}(6)(4) \times 10 = 120

ft³. Total volume =

300 + 120 = 420

ft³. (A) uses an incorrect triangular area calculation. (C) miscalculates the rectangular prism volume. (D) forgets the

\frac{1}{2}

factor in the triangular prism volume calculation.

Question 10

A rectangular room is 14 feet long, 10 feet wide, and 8 feet high. The four walls and the ceiling are to be painted. What is the total area to be painted in square feet?

192 sq ft
384 sq ft
524 sq ft (correct answer)
664 sq ft

Explanation: First, calculate the area of the four walls (the lateral surface area). The perimeter of the floor is 2(14 + 10) = 2(24) = 48 feet. The area of the walls is the perimeter multiplied by the height: 48 ft × 8 ft = 384 sq ft. Next, calculate the area of the ceiling, which is the same as the area of the floor: 14 ft × 10 ft = 140 sq ft. The total area to be painted is the sum of the walls' area and the ceiling's area: 384 sq ft + 140 sq ft = 524 sq ft.

Question 11

A cylindrical can has a volume of 72π cubic centimeters and a height of 8 centimeters. What is the circumference of its base?

3π cm
6π cm (correct answer)
9π cm
12π cm

Explanation: The volume of a cylinder is V = πr²h. We are given V = 72π and h = 8. So, 72π = πr²(8). Divide both sides by 8π: 9 = r². Taking the square root gives the radius r = 3 cm. The question asks for the circumference of the base, which is C = 2πr. Substituting r = 3, we get C = 2π(3) = 6π centimeters.

Question 12

The diagonal of a square is 10 inches long. What is the area of the square in square inches?

20 sq in
50 sq in (correct answer)
100 sq in
200 sq in

Explanation: Let the side length of the square be 's'. The diagonal, 'd', forms a right triangle with two sides of the square. By the Pythagorean theorem, s² + s² = d². This simplifies to 2s² = d². We are given d = 10, so 2s² = 10² = 100. The area of the square is A = s². From the equation 2s² = 100, we can solve for s² by dividing by 2: s² = 50. Therefore, the area of the square is 50 square inches.

Question 13

A company manufactures cylindrical cans. The height of a can is 10 cm and the diameter of its base is 8 cm. What is the total surface area of the can in terms of π?

80π sq cm
96π sq cm
112π sq cm (correct answer)
144π sq cm

Explanation: The total surface area of a cylinder is SA = 2πr² + 2πrh. The diameter is 8 cm, so the radius r = 4 cm. The height h = 10 cm. The area of the two circular bases is 2πr² = 2π(4)² = 2π(16) = 32π sq cm. The lateral surface area (the side) is 2πrh = 2π(4)(10) = 80π sq cm. The total surface area is the sum of these two parts: 32π + 80π = 112π sq cm.

Question 14

The area of a circle is 144π square units. What is the circumference of the circle?

12π units
24π units (correct answer)
36π units
144π units

Explanation: The formula for the area of a circle is A = πr². We are given A = 144π. So, 144π = πr². Dividing both sides by π gives r² = 144. Taking the square root of both sides gives r = 12 units. The formula for the circumference of a circle is C = 2πr. Substituting r = 12, we get C = 2π(12) = 24π units.

Question 15

A rectangular sandbox has a length of 8 feet and a width of 6 feet. It is filled with sand to a uniform depth of 9 inches. What is the volume of sand in the sandbox in cubic feet?

36 cu ft (correct answer)
48 cu ft
72 cu ft
432 cu ft

Explanation: To find the volume in cubic feet, all dimensions must be in feet. The length is 8 ft and the width is 6 ft. The depth is 9 inches. To convert inches to feet, divide by 12: 9 inches = 9/12 feet = 0.75 feet. Now, calculate the volume: V = length × width × depth = 8 ft × 6 ft × 0.75 ft = 48 × 0.75 = 36 cubic feet.

Question 16

The ratio of the volumes of two cubes is 8:27. What is the ratio of their surface areas?

2:3
4:9 (correct answer)
8:27
16:81

Explanation: Let the side lengths of the two cubes be s₁ and s₂. The ratio of their volumes is (s₁)³ / (s₂)³ = 8/27. Taking the cube root of both sides gives s₁/s₂ = 2/3. The surface area of a cube is 6s². The ratio of their surface areas is (6(s₁)²) / (6(s₂)²) = (s₁/s₂)² . Since s₁/s₂ = 2/3, the ratio of the surface areas is (2/3)² = 4/9.

Question 17

A parallelogram has an area of 72 square centimeters and a base of 9 centimeters. A triangle has the same area and the same base as the parallelogram. What is the height of the triangle?

4 cm
8 cm
16 cm (correct answer)
18 cm

Explanation: This is a multi-step problem. First, find the height of the parallelogram. Area = base × height. 72 = 9 × h_para. So, h_para = 8 cm. Now, consider the triangle. The triangle's area is 72 sq cm and its base is 9 cm. The area formula for a triangle is A = (1/2)bh. 72 = (1/2) × 9 × h_tri. Multiply both sides by 2: 144 = 9 × h_tri. Divide by 9: h_tri = 16 cm. The height of the triangle is 16 cm.

Question 18

A semi-circular protractor has a diameter of 10 centimeters. What is the perimeter of the protractor in centimeters? (Use π ≈ 3.14)

15.70 cm
25.70 cm (correct answer)
31.40 cm
41.40 cm

Explanation: The perimeter of the protractor consists of two parts: the curved semi-circular arc and the straight diameter. The length of the full circumference of a circle with diameter 10 cm is C = πd = 3.14 × 10 = 31.4 cm. The length of the semi-circular arc is half of this, which is 31.4 / 2 = 15.7 cm. The length of the straight edge is the diameter, which is 10 cm. The total perimeter is the sum of the arc and the diameter: 15.7 cm + 10 cm = 25.70 cm.

Question 19

The length of a rectangle is increased by 20% and its width is decreased by 10%. What is the percent change in the area of the rectangle?

8% increase (correct answer)
10% increase
10% decrease
12% increase

Explanation: Let the original length be L and the original width be W. The original area is A = LW. The new length is L' = L + 0.20L = 1.2L. The new width is W' = W - 0.10W = 0.9W. The new area is A' = L'W' = (1.2L)(0.9W) = 1.08LW. The change in area is A' - A = 1.08LW - LW = 0.08LW. The percent change is (change / original) × 100 = (0.08LW / LW) × 100 = 0.08 × 100 = 8%. Since the result is positive, it is an 8% increase.

Question 20

A solid metal cube with a side length of 6 inches is melted down and recast into a solid cylinder with a radius of 3 inches. What is the height of the cylinder?

8/π inches
12/π inches
24/π inches (correct answer)
36/π inches

Explanation: The volume of the material remains the same. First, calculate the volume of the cube: V_cube = s³ = 6³ = 216 cubic inches. Next, use the volume formula for the cylinder, V_cylinder = πr²h. We know the volume must be 216 and the radius is 3. So, 216 = π(3)²h = 9πh. To solve for the height h, divide both sides by 9π: h = 216 / (9π) = 24/π inches.