Exercises: Understand radian measure - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

The circumference of a circle with radius $r$ is:

A.

$\pi r^2$

B.

$2\pi r$

C.

$\frac{r}{2\pi}$

D.

$\pi r$

2.

How many degrees are in a full rotation?

A.

90°

B.

180°

C.

270°

D.

360°

3.

A circle has a central angle of 90°. What fraction of the full circle does this angle sweep out?

A.

$\frac{1}{6}$

B.

$\frac{1}{4}$

C.

$\frac{1}{3}$

D.

$\frac{1}{2}$

B

Fluency Practice

Convert each angle as directed. Express radian answers as exact values in terms of $\pi$ .

1.

Convert 60° to radians. Express your answer as a fraction in terms of $\pi$ .

2.

Convert 150° to radians. Express your answer as a fraction in terms of $\pi$ .

3.

Convert $\frac{\pi}{4}$ radians to degrees.

4.

Convert $\frac{5\pi}{6}$ radians to degrees.

5.

Which radian measure is equivalent to 270°?

A.

$\pi$ radians

B.

$\frac{3\pi}{2}$ radians

C.

$\frac{\pi}{3}$ radians

D.

$2\pi$ radians

C

Mixed Practice

These problems test the same skills in different ways.

D

Word Problems

Apply your knowledge of radian measure to each scenario.

$Circle with a radius of 15 cm and a central angle of pi/3 radians, with arc length s labeled as unknown.$

1.

A clock's minute hand is 15 cm long. It sweeps through an angle of $\frac{\pi}{3}$ radians.

How many centimeters does the tip of the minute hand travel? Use the arc length formula $s = r\theta$ .

2.

A pizza is cut into 8 equal slices. The pizza has a radius of 12 inches.

What is the central angle of each slice in radians? Express your answer as a fraction in terms of $\pi$ .

3.

A sector of a circle has radius 6 cm and central angle $\frac{2\pi}{3}$ radians.

1.

Find the arc length of the sector. Use $s = r\theta$ .

2.

What is the degree measure of the central angle $\frac{2\pi}{3}$ ?

E

Error Analysis

Each problem shows a student's incorrect reasoning. Identify the mistake.

1.

Jordan says: "1 radian is exactly 60 degrees, because there are about 6 radians in a full circle, and $360\degree \div 6 = 60\degree$ ."

What is wrong with Jordan's reasoning?

A.

Jordan is correct — 1 radian is exactly 60°.

B.

Jordan rounded the number of radians in a full circle. The exact value is $2\pi \approx 6.283$ , not 6, so $1 \text{ rad} = \frac{180\degree}{\pi} \approx 57.3\degree$ , not 60°.

C.

Jordan should have divided 360° by $2\pi$ , but the answer would still be 60°.

D.

The number of radians in a full circle is 12, not 6, so 1 radian = 30°.

2.

Alex writes: "Since $\pi = 180$ , I can replace $\pi$ with 180 anywhere it appears in a formula. So the circumference formula becomes $C = 2(180)r = 360r$ ."

What fundamental error has Alex made?

A.

Alex has the right idea — $\pi$ and 180 are interchangeable in trigonometry.

B.

Alex confused a unit conversion with a numerical equality. The correct statement is " $\pi$ radians $= 180\degree$ " — $\pi$ the number is approximately 3.14159, not 180.

C.

Alex should have written $\pi = 360$ instead.

D.

Alex is correct only for angles less than 90°.

F

Challenge / Extension

These problems extend the core skill. Try them after completing the rest of the set.

1.

A circle has radius 7 cm. An arc on this circle has length 14 cm.

Find the radian measure of the central angle subtending the arc. (Hint: use $\theta = \frac{s}{r}$ .)

2.