Exercises: Circumference and area - ACT Math

A

Recall / Warm-Up

1.

A circle has a diameter of 14 cm. What is the radius?

A.

7 cm

B.

14 cm

C.

28 cm

D.

3.5 cm

2.

What is the perimeter of a square with side length 8 cm?

A.

16 cm

B.

32 cm

C.

64 cm

D.

24 cm

3.

What is $7^2$ ?

B

Fluency Practice

1.

A circle has radius 5 cm. What is the circumference? Express your answer in terms of $\pi$ .

2.

A circle has diameter 18 in. What is the circumference in terms of $\pi$ ?

A.

$9\pi$ in

B.

$18\pi$ in

C.

$36\pi$ in

D.

18 in

3.

A circle has radius 6 cm. What is the area? Express your answer in terms of $\pi$ .

4.

A circle has diameter 10 m. What is its area?

A.

$100\pi$ m²

B.

$25\pi$ m²

C.

$10\pi$ m²

D.

$50\pi$ m²

5.

A circle has radius 9 cm. What is the circumference? Round to the nearest tenth. (Use $\pi \approx 3.14159$ .)

C

Varied Practice

1.

A circle has circumference $30\pi$ ft. What is the radius?

A.

30 ft

B.

15 ft

C.

60 ft

D.

$\frac{30}{\pi}$ ft

2.

A circle has area $49\pi$ in². What is the radius in inches?

3.

A circle has circumference $20\pi$ cm. What is the area of the circle in terms of $\pi$ ?

A.

$100\pi$ cm²

B.

$400\pi$ cm²

C.

$200\pi$ cm²

D.

$40\pi$ cm²

4.

A circle has radius 4 cm. The circumference is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ cm and the area is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ cm². Express both answers in terms of $\pi$ .

circumference:

area:

D

Word Problems / Application

1.

A circle with radius 5 is inscribed in a square (the circle touches all four sides of the square).

What is the area of the region between the square and the circle?

A.

$100 - 25\pi$

B.

$25\pi - 100$

C.

$100 - 10\pi$

D.

$400 - 25\pi$

2.

An annulus (ring) is formed by two concentric circles. The outer circle has radius 8 cm and the inner circle has radius 5 cm.

What is the area of the annulus? Express your answer in terms of $\pi$ .

3.

A semicircle has radius 4 cm.

1.

What is the area of the semicircle? Express your answer in terms of $\pi$ .

2.

What is the perimeter of the semicircle in terms of $\pi$ ?

A.

$4\pi$ cm

B.

$4\pi + 8$ cm

C.

$8\pi$ cm

D.

$4\pi + 4$ cm

4.

A semicircular window has a perimeter of $(6\pi + 12)$ inches.

What is the area of the window? Express your answer in terms of $\pi$ .

E

Error Analysis

1.

Marcus solved this problem:

"A circle has diameter 12 cm. Find the area."

Marcus's work:

$A = \pi(12)^2$
$A = 144\pi$ cm²

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

A.

Marcus used the diameter instead of the radius. The correct area is $36\pi$ cm².

B.

Marcus forgot to divide by 2. The correct area is $72\pi$ cm².

C.

Marcus's answer of $144\pi$ cm² is correct.

D.

Marcus should have used $2\pi r$ . The correct area is $12\pi$ cm².

2.

Taylor solved this problem:

"A semicircle has radius 10 cm. Find the perimeter."

Taylor's work:

Half the circumference: $\frac{1}{2}(2\pi)(10) = 10\pi$
Perimeter $= 10\pi \approx 31.4$ cm

What did Taylor forget, and what is the correct perimeter?

A.

Taylor forgot to add the diameter. The correct perimeter is $10\pi + 20$ cm.

B.

Taylor forgot to add the radius. The correct perimeter is $10\pi + 10$ cm.

C.

Taylor's answer of $10\pi$ cm is correct.

D.

Taylor should have used the full circumference. The correct perimeter is $20\pi$ cm.

F

Challenge / Extension

1.

A quarter circle has radius 6 cm. What is its area? Express your answer in terms of $\pi$ .

2.

A square with side length 6 is inscribed in a circle (all four vertices touch the circle).

Find the area of the shaded region between the circle and the square. Express your answer in terms of $\pi$ .

3.