Radian Measure Examples: 42 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Radian Measure.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An angle measure defined by the arc length subtended on a unit circle: one radian is the angle that subtends an arc equal in length to the radius.

Imagine wrapping the radius of a circle along its edge like a piece of string. The angle you've swept out is exactly 1 radian. Since the full circumference is $2\pi r$ , a full turn is $2\pi$ radians. Radians measure angles in terms of the circle itself, which is why they're the natural unit for calculus and physics—no arbitrary conversion factor like $360$ is needed.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: One radian is the angle whose arc equals one radius, so a full turn is $2\pi$ .

Common stuck point: The procedure for radian measure is the easy part; the trap is plugging a degree value into a calculator set to radians. Asking "Is the angle measured by arc-lengths-of-radius around the circle rather than by a slice of 360?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the angle measured by arc-lengths-of-radius around the circle rather than by a slice of 360?

Worked Examples

Example 1

easy

Convert

135°

to radians and

\dfrac{5\pi}{6}

radians to degrees. Show the conversion steps.

Answer

135° = \dfrac{3\pi}{4}

rad;

\dfrac{5\pi}{6}

rad

= 150°

First step

135°

to radians: multiply by

\frac{\pi}{180}

135\times\frac{\pi}{180}=\frac{135\pi}{180}=\frac{3\pi}{4}

rad.

Full solution

2
$\frac{5\pi}{6}$ rad to degrees: multiply by $\frac{180}{\pi}$ . $\frac{5\pi}{6}\times\frac{180}{\pi}=\frac{5\times180}{6}=\frac{900}{6}=150°$ .
3
Memory aid: $\pi$ rad $= 180°$ ; to go to radians multiply by $\pi/180$ ; to go to degrees multiply by $180/\pi$ .

Radians and degrees both measure angles;

\pi

radians

= 180°

is the fundamental conversion. Radians are preferred in calculus because they make derivative formulas for trig functions clean (no extra factors).

Example 2

medium

A wheel of radius

5

cm rotates through an angle of

2.4

radians. Find the arc length and the area of the sector swept.

Example 3

medium

A central angle of

1.5

radians is subtended on a circle of radius

10

cm. Find the arc length and the area of the sector.

Example 4

medium

A pendulum swings through an angle of

36^\circ

. Express that angle in radians (exact).

Example 5

medium

Convert

\dfrac{11\pi}{12}

radians to degrees.

Example 6

hard

A circular sector has area

20\pi

^2

and central angle

\dfrac{2\pi}{5}

rad. Find the radius.

Example 7

hard

A satellite orbits Earth at altitude giving orbit radius

7000

km. It travels

1400

km along its orbit. Through what angle (radians) has it moved? Convert to degrees.

Example 8

hard

A bicycle wheel of radius

35

cm rotates at

120

revolutions per minute. Find the linear speed of a point on the rim in m/s.

Example 9

challenge

A sector has perimeter

24

cm (two radii plus the arc). For what central angle

\theta

(in radians) is the sector area a maximum, and what is that maximum area?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Convert: (a)

60°

to radians, (b)

270°

to radians, (c)

\dfrac{\pi}{3}

to degrees, (d)

\dfrac{7\pi}{6}

to degrees.

Example 2

medium

A clock's minute hand is

315^\circ

to radians.

Example 27

medium

Find the area of a sector with radius

6

and central angle

\dfrac{\pi}{3}

Example 28

medium

A wheel of radius

0.5

m rolls without slipping. Through how many radians does it turn after traveling

4

Example 29

medium

A circle has radius

9

. Find the arc length subtended by an angle of

80^\circ

Example 30

medium

Find the radius of a circle if a

2

-rad central angle subtends an arc of length

9

Example 31

hard

A circular track has radius

50

m. A runner runs

200

m along the track. Through what angle (in radians) does she sweep around the center?

Example 32

hard

A sector of a circle has radius

r

and the arc length equals the radius. What is the central angle in radians?

Example 33

hard

On a circle of radius

12

, two radii form a central angle of

50^\circ

. Find the exact area of the sector.

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Related Concepts

Unit Circle Pi (π)Trigonometric Function Graphs Arc Length

Background Knowledge

These ideas may be useful before you work through the harder examples.

unit circlepi