Exercises: Arc length and sector area - Digital SAT Math

A

Recall / Warm-Up

1.

What is the circumference of a circle with radius $9$ cm? Express your answer in terms of $\pi$ .

2.

What is the area of a circle with radius $8$ inches? Express your answer in terms of $\pi$ .

3.

Convert $\dfrac{\pi}{3}$ radians to degrees.

A.

$30\degree$

B.

$45\degree$

C.

$60\degree$

D.

$90\degree$

B

Fluency Practice

1.

Find the arc length of a central angle of $120\degree$ in a circle of radius $9$ cm. Express your answer in terms of $\pi$ .

2.

Find the arc length of a central angle of $\dfrac{2\pi}{3}$ radians in a circle of radius $9$ cm. Express your answer in terms of $\pi$ .

3.

Find the area of a sector with central angle $90\degree$ and radius $8$ inches. Express your answer in terms of $\pi$ .

4.

Find the area of a sector with central angle $\dfrac{\pi}{2}$ radians and radius $8$ inches. Express your answer in terms of $\pi$ .

C

Varied Practice

1.

A circle has radius $6$ and a central angle of $60\degree$ . Which expression gives the arc length?

A.

$\dfrac{60}{360} \cdot \pi \cdot 6^2$

B.

$\dfrac{60}{360} \cdot 2\pi \cdot 6$

C.

$6 \cdot 60$

D.

$\pi \cdot 6^2 \cdot 60$

2.

A circle of radius $5$ has a sector with arc length $\dfrac{5\pi}{3}$ . What is the central angle in degrees?

3.

A sector has central angle $\dfrac{\pi}{4}$ radians and area $18\pi$ . What is the radius?

4.

Convert $150\degree$ to radians.

A.

$\dfrac{5\pi}{4}$

B.

$\dfrac{5\pi}{6}$

C.

$\dfrac{3\pi}{4}$

D.

$\dfrac{\pi}{3}$

5.

Using the radian arc-length formula: arc length $= r \cdot \theta$ . For a circle of radius $10$ and central angle $\dfrac{3\pi}{4}$ , the arc length is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $\cdot$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $=$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

radius:

angle (radians):

arc length:

D

Word Problems / Application

1.

A clock has an hour hand that is $6$ inches long. The hand moves from 12 to 4, sweeping through $120\degree$ .

What arc length does the tip of the hour hand travel? Express your answer in terms of $\pi$ .

2.

A circular pizza has radius $12$ inches. A slice is cut with a central angle of $45\degree$ .

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

3.

A windshield wiper sweeps through an angle of $\dfrac{2\pi}{3}$ radians. The wiper blade is $15$ inches long and extends from $5$ inches to $20$ inches from the pivot point.

1.

What is the arc length swept by the tip of the wiper blade (at $20$ inches from the pivot)? Express your answer in terms of $\pi$ .

2.

What is the area swept by the wiper blade (the annular sector from radius $5$ to radius $20$ )? Express your answer in terms of $\pi$ .

4.

A sprinkler rotates through $270\degree$ and waters a circular sector of radius $8$ meters.

What is the area watered by the sprinkler? Express your answer in terms of $\pi$ .

E

Error Analysis

1.

Alex computed the arc length for a central angle of $\\dfrac{\\pi}{3}$ radians and radius $12$ :

Alex's work:
$\\text{arc} = \\frac{(\\pi/3)}{360} \\cdot 2\\pi \\cdot 12 = \\frac{\\pi}{1080} \\cdot 24\\pi \\approx 0.069$

What mistake did Alex make?

A.

Alex used the wrong formula for arc length.

B.

Alex made an arithmetic error in simplifying the fraction.

C.

Alex plugged a radian angle into the degree formula. The correct method uses $s = r\theta = 12 \cdot \dfrac{\pi}{3} = 4\pi$ .

D.

Alex should have used the sector area formula instead of the arc length formula.

2.

Jordan computed the sector area for a central angle of $90\degree$ and radius $6$ :

Jordan's work:
$\\text{Area} = \\frac{90}{360} \\cdot 2\\pi \\cdot 6 = \\frac{1}{4} \\cdot 12\\pi = 3\\pi$

What two errors did Jordan make?

A.

Jordan used the arc length formula instead of the sector area formula, and also forgot to square the radius.

B.

Jordan made an error in simplifying $\dfrac{90}{360}$ to $\dfrac{1}{4}$ .

C.

Jordan should have converted $90\degree$ to radians before computing.

D.

Jordan correctly computed the area.

F

Challenge / Extension

1.

A sector has a perimeter of $30 + 5\pi$ cm. The two straight sides (radii) each have length $15$ cm. Find the central angle of the sector in degrees.

2.

Two circles are concentric (same center). The inner circle has radius $4$ and the outer circle has radius $10$ . A central angle of $\dfrac{\pi}{4}$ radians defines an annular sector — the region between the two circles within that angle.

1.

Find the area of the annular sector. Express your answer in terms of $\pi$ .

2.