Exercises: Surface area of prisms, cylinders, pyramids, cones, spheres - ACT Math

A

Recall / Warm-Up

1.

What is the area of a rectangle with length 9 cm and width 7 cm?

A.

32 cm²

B.

63 cm²

C.

16 cm²

D.

126 cm²

2.

What is the circumference of a circle with radius 5 cm? Leave your answer in terms of $\pi$ .

A.

$5\pi$ cm

B.

$10\pi$ cm

C.

$25\pi$ cm

D.

$20\pi$ cm

3.

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

A.

14

B.

10

C.

7

D.

48

B

Fluency Practice

1.

A rectangular prism has length 10 cm, width 6 cm, and height 4 cm. What is the total surface area in square centimeters?

2.

A cylinder has radius 3 cm and height 8 cm. What is the lateral surface area? Express your answer in terms of $\pi$ .

A.

$24\pi$ cm²

B.

$48\pi$ cm²

C.

$66\pi$ cm²

D.

$72\pi$ cm²

3.

A cylinder has radius 5 cm and height 12 cm. What is the total surface area? Express your answer in terms of $\pi$ .

4.

A square pyramid has base edge 10 m and slant height 13 m. What is the total surface area?

A.

260 m²

B.

360 m²

C.

160 m²

D.

560 m²

5.

A sphere has radius 7 cm. What is the surface area? Express your answer in terms of $\pi$ .

C

Varied Practice

D

Word Problems / Application

1.

A storage tank is a closed cylinder with radius 3 m and height 8 m. The outside of the tank (top, bottom, and curved side) needs to be painted.

What is the total area to be painted? Express your answer in terms of $\pi$ .

2.

A tent is shaped like a cone with a circular base. The base has a radius of 8 ft and the tent has a perpendicular height of 6 ft. The tent floor is the circular base.

1.

What is the slant height of the tent in feet?

2.

What is the total surface area of the tent (including the floor)? Express your answer in terms of $\pi$ .

3.

A rectangular box (no lid) has length 12 in, width 8 in, and height 5 in. The outside of the box needs to be wrapped in decorative paper.

What is the total area of paper needed in square inches?

E

Error Analysis

1.

Alex solved this problem:

"A cone has radius 5 cm and height 12 cm. Find the lateral surface area."

Alex's work:

$\text{LSA} = \pi r h$
$\text{LSA} = \pi(5)(12) = 60\pi$ cm²

What error did Alex make, and what is the correct lateral surface area?

A.

Alex used the height instead of the slant height. The correct LSA is $65\pi$ cm².

B.

Alex forgot the factor of 2. The correct LSA is $120\pi$ cm².

C.

Alex's answer of $60\pi$ cm² is correct.

D.

Alex should have used $\frac{1}{2}\pi r h$ . The correct LSA is $30\pi$ cm².

2.

Taylor solved this problem:

"A sphere has radius 6 cm. Find the surface area."

Taylor's work:

$SA = \frac{4}{3}\pi r^3$
$SA = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi$ cm²

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

A.

Taylor used the volume formula instead of the surface area formula. The correct SA is $144\pi$ cm².

B.

Taylor forgot to divide by 3. The correct SA is $96\pi$ cm².

C.

Taylor's answer of $288\pi$ cm² is correct.

D.

Taylor should have used $2\pi r^2$ . The correct SA is $72\pi$ cm².

F

Challenge / Extension

1.

A solid is formed by placing a hemisphere (half-sphere) on top of a cylinder. Both have radius 4 cm, and the cylinder has height 10 cm. What is the total surface area of the solid? Express your answer in terms of $\pi$ .

2.