Exercises: Surface area of 3D solids (prisms, cylinders, cones, pyramids, spheres) - Digital SAT Math

A

Recall / Warm-Up

1.

What is the area of a rectangle with length 9 cm and width 4 cm?

A.

13 cm²

B.

26 cm²

C.

36 cm²

D.

72 cm²

2.

A rectangular box has length 5 cm, width 3 cm, and height 2 cm. What is the area, in square centimeters, of the face that measures 5 cm by 3 cm?

3.

A square has side length 7 inches. What is its area, expressed with the correct units?

A.

14 inches

B.

28 inches

C.

49 inches

D.

49 in²

B

Fluency Practice

$Rectangular prism with length 8 cm, width 5 cm, height 3 cm labeled on each visible edge.$

1.

A rectangular prism has length 8 cm, width 5 cm, and height 3 cm. What is the total surface area, in square centimeters?

2.

A rectangular prism has length 10 m, width 2.5 m, and height 4 m. What is the total surface area, in square meters?

$Cylinder net showing two circles of radius 3 cm and a rectangle of height 8 cm whose width equals the circumference 6π cm.$

3.

A cylinder has radius 3 cm and height 8 cm. What is its total surface area? Express your answer in terms of π (e.g., write 66pi).

4.

A cylinder has diameter 10 cm and height 6 cm. What is its total surface area? Express your answer in terms of π.

5.

A square pyramid has a base edge of 6 m and a slant height of 5 m. What is its total surface area, in square meters?

6.

A cone has radius 4 cm and slant height 9 cm. What is its total surface area? Express your answer in terms of π.

C

Varied Practice

D

Word Problems / Application

1.

A cylindrical water tank has a diameter of 8 ft and a height of 15 ft. The exterior of the tank needs to be painted. Only the curved side is painted — the circular top and bottom are not accessible. Paint costs $2.50 per square foot.

What is the total cost, in dollars, to paint the exterior of the tank? Use π ≈ 3.14 and round to the nearest dollar.

2.

A manufacturer makes gift boxes in the shape of rectangular prisms. Each box is 20 cm long, 15 cm wide, and 10 cm tall. The boxes are wrapped with no overlapping paper.

What is the minimum amount of wrapping paper, in square centimeters, needed for one box?

A.

1,300 cm²

B.

1,500 cm²

C.

3,000 cm²

D.

6,000 cm²

3.

A sports equipment company makes foam balls shaped like spheres. One size of ball has a radius of 5 cm. The company sells foam material priced by surface area.

1.

What is the surface area of the ball in terms of π? Write your answer in the form kπ cm².

2.

If the foam material costs $0.04 per cm², what is the cost of foam for one ball? Use π ≈ 3.14 and round to the nearest cent.

4.

A pyramid-shaped roof ornament has a square base with edge length 12 in and a slant height of 10 in. The base is attached to the building and is not painted. Only the four triangular faces need to be painted.

What is the total painted surface area of the ornament, in square inches?

E

Error Analysis

$Side-by-side contrast: Maya incorrectly used vertical height h = 4 m (left) versus the correct slant height l = 5 m derived from Pythagorean theorem (right).$

1.

Maya solved this problem:

"A square pyramid has a base edge of 6 m and a vertical height of 4 m. Find the total surface area."

Maya's work:

Base area: $6^2 = 36$ m²
Base perimeter: $4 \times 6 = 24$ m
Lateral SA: $(1/2)(24)(4) = 48$ m²
Total SA = $36 + 48 =$ 84 m²

What error did Maya make, and what is the correct total surface area?

A.

Maya used the vertical height (4 m) instead of the slant height. The slant height is 5 m. The correct total SA is 96 m².

B.

Maya forgot to double the base area. The correct total SA is 120 m².

C.

Maya's answer of 84 m² is correct.

D.

Maya used the wrong perimeter formula; the correct perimeter is 6 m.

$Side-by-side cylinders: left shows Luca missing both bases (dashed ellipses), right shows correct total SA including both solid circular bases.$

2.

Luca solved this problem:

"A cylinder has radius 6 cm and height 14 cm. Find its total surface area."

Luca's work:

Lateral SA = $2\pi(6)(14) = 168\pi$ cm²
Total SA = $168\pi$ cm²

What error did Luca make, and what is the correct total surface area in terms of π?

A.

Luca used the wrong lateral SA formula. The correct lateral SA is $\pi(6)^2(14)$ .

B.

Luca forgot to include the two circular bases. The correct total SA is $168\pi + 72\pi = 240\pi$ cm².

C.

Luca's answer is correct; $168\pi$ cm² is the total surface area.

D.

Luca should have used the diameter (12 cm) instead of the radius. The correct SA is $336\pi$ cm².

F

Challenge / Extension

1.

A composite solid is formed by placing a hemisphere (half-sphere) on top of a cylinder. The cylinder has radius 4 cm and height 10 cm. The hemisphere has the same radius (4 cm) and sits flush on top of the cylinder, replacing the top circular face. Include: the bottom circular base of the cylinder, the lateral surface of the cylinder, and the curved surface of the hemisphere. Do not include the flat circular interface between the hemisphere and the cylinder. What is the total surface area of the composite solid, in terms of π?

2.