Radians Examples in Math

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

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

Concept Recap

A radian is an angle measurement defined by the arc length it subtends on a unit circle: one radian is the angle at which the arc length equals the radius. A full circle is $2\pi$ radians (about 6.28 radians), making radians the natural unit for trigonometry and calculus.

It ties angle directly to the circle’s geometry instead of degree counting.

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: A radian measures an angle by arc length per radius, so a full circle is $2\pi$ radians.

Common stuck point: The procedure for radians is the easy part; the trap is leaving the calculator in degree mode for radian work (or vice versa). Asking "Is the angle measured by arc-length-over-radius (so a full circle is $2\pi$ , not $360$ )?" 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-length-over-radius (so a full circle is $2\pi$ , not $360$ )?

Worked Examples

Example 1

easy

Convert

150°

to radians.

Answer

\frac{5\pi}{6} \text{ radians}

First step

Use the conversion factor:

180° = \pi

radians.

Full solution

2
Multiply: $150° \times \frac{\pi}{180°} = \frac{150\pi}{180}$ .
3
Simplify: $\frac{150\pi}{180} = \frac{5\pi}{6}$ .

Radians measure angles by the arc length subtended on a unit circle. Since the full circle has circumference

2\pi

, a full rotation is

2\pi

radians

= 360°

. The conversion factor

\frac{\pi}{180}

converts degrees to radians.

Example 2

medium

Find the arc length of a sector with radius

10

cm and central angle

\frac{3\pi}{4}

radians.

Example 3

medium

Find the arc length on a circle of radius

8

cm subtended by an angle of

\frac{2\pi}{5}

radians.

Example 4

medium

Convert

\frac{11\pi}{6}

radians to degrees and find a coterminal positive angle less than

2\pi

Example 5

medium

Convert

1.5

radians to degrees and decide whether the angle is acute, right, obtuse, or reflex.

Example 6

medium

A car tire of radius

0.35

m makes

5

full rotations. What total distance does a point on the rim travel?

Example 7

hard

Two circles share a center. Circle A has radius

4

and circle B has radius

10

. A central angle of

\pi/3

subtends arc

s_A

in A and

s_B

in B. Find the difference

s_B - s_A

Example 8

hard

Compute the linear speed at the equator due to Earth's rotation if Earth's radius is

6378

km and it rotates once every

24

hours.

Example 9

challenge

Show that for small

\theta

in radians,

\sin\theta \approx \theta

. Use it to estimate

\sin(0.05)

Practice Problems

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

Example 1

medium

Convert

\frac{7\pi}{4}

radians to degrees and identify which quadrant this angle is in.

Example 2

hard

A wheel of radius

30

cm rotates at

120

rpm (revolutions per minute). Find the angular velocity in radians per second and the linear speed of a point on the rim.

Example 3

easy

Convert

180°

to radians.

Example 4

easy

Convert

90°

to radians.

Example 5

easy

Convert

45°

to radians.

Example 6

easy

Convert

360°

to radians.

Example 7

easy

Convert

\pi

radians to degrees.

Example 8

easy

Convert

\frac{\pi}{3}

radians to degrees.

Example 9

easy

Convert

30°

to radians.

Example 10

easy

How many radians are in a full circle?

Example 11

medium

Convert

\frac{3\pi}{4}

radians to degrees.

Example 12

medium

Convert

120°

to radians.

Example 13

medium

Find the arc length on a circle of radius

5

subtended by an angle of

\frac{\pi}{3}

radians.

Example 14

medium

Evaluate

\sin\left(\frac{\pi}{2}\right)

Example 15

medium

Convert

1

radian to degrees (approximate to the nearest degree).

Example 16

medium

Find the area of a sector with radius

4

and central angle

\frac{\pi}{2}

Example 17

medium

Convert

270°

to radians.

Example 18

medium

A wheel turns through

4\pi

radians. How many full revolutions is that?

Example 19

challenge

A central angle subtends an arc of length

6

on a circle of radius

4

. Find the angle in radians.

Example 20

challenge

Evaluate

\cos\left(\frac{2\pi}{3}\right)

Example 21

medium

Convert

225

degrees to radians.

Example 22

challenge

An angle measures

\frac{5\pi}{6}

radians. Express it in degrees and state its reference angle in degrees.

Example 23

easy

Convert

270^\circ

to radians.

Example 24

easy

Convert

\frac{2\pi}{3}

radians to degrees.

Example 25

easy

Convert

60^\circ

to radians.

Example 26

easy

Convert

\frac{5\pi}{4}

radians to degrees.

Example 27

easy

In a unit circle, what arc length is subtended by an angle of

\pi/4

radians?

Example 28

medium

Find the area of a sector with radius

6

and central angle

\pi/3

radians.

Example 29

medium

A central angle of

\frac{\pi}{2}

radians subtends an arc of length

5

cm. What is the radius?

Example 30

medium

In which quadrant does an angle of

\frac{4\pi}{3}

radians lie?

Example 31

medium

Compute

\sin\left(\frac{\pi}{6}\right) + \cos\left(\frac{\pi}{3}\right)

Example 32

medium

A pendulum swings through an arc of

0.8

m on a string of length

2

m. Find the angle swept in radians.

Example 33

hard

Find the area of a sector with radius

12

cm and arc length

9

cm.

Example 34

hard

A wheel spins at

300

rpm. Express its angular speed in radians per second.

Example 35

hard

Find

\tan\left(\frac{5\pi}{6}\right)

Example 36

hard

Solve

2 \sin x = 1

for

x \in [0, 2\pi)

Example 37

hard

Find the area of the region cut off by a chord in a unit circle if the chord subtends a central angle of

\pi/2

Example 38

challenge

On a circle of radius

r

, an inscribed angle subtends an arc of length

\pi r / 3

. Find the inscribed angle in radians.

← Back to Radians Formula Explained Practice Mode

Related Concepts

Pi (π)Arc Length Unit Circle

Background Knowledge

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

piarc lengthunit circle