Radian Measure Examples in Math

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 2πr2\pi r, a full turn is 2π2\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 360360 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 2π2\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°135° to radians and 5π6\dfrac{5\pi}{6} radians to degrees. Show the conversion steps.

Answer

135°=3π4135° = \dfrac{3\pi}{4} rad; 5π6\dfrac{5\pi}{6} rad =150°= 150°

First step

1
135°135° to radians: multiply by π180\frac{\pi}{180}. 135×π180=135π180=3π4135\times\frac{\pi}{180}=\frac{135\pi}{180}=\frac{3\pi}{4} rad.

Full solution

  1. 2
    5π6\frac{5\pi}{6} rad to degrees: multiply by 180π\frac{180}{\pi}. 5π6×180π=5×1806=9006=150°\frac{5\pi}{6}\times\frac{180}{\pi}=\frac{5\times180}{6}=\frac{900}{6}=150°.
  2. 3
    Memory aid: π\pi rad =180°= 180°; to go to radians multiply by π/180\pi/180; to go to degrees multiply by 180/π180/\pi.
Radians and degrees both measure angles; π\pi radians =180°= 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 55 cm rotates through an angle of 2.42.4 radians. Find the arc length and the area of the sector swept.

Example 3

medium
A central angle of 1.51.5 radians is subtended on a circle of radius 1010 cm. Find the arc length and the area of the sector.

Example 4

medium
A pendulum swings through an angle of 36∘36^\circ. Express that angle in radians (exact).

Example 5

medium
Convert 11π12\dfrac{11\pi}{12} radians to degrees.

Example 6

hard
A circular sector has area 20π20\pi cm2^2 and central angle 2π5\dfrac{2\pi}{5} rad. Find the radius.

Example 7

hard
A satellite orbits Earth at altitude giving orbit radius 70007000 km. It travels 14001400 km along its orbit. Through what angle (radians) has it moved? Convert to degrees.

Example 8

hard
A bicycle wheel of radius 3535 cm rotates at 120120 revolutions per minute. Find the linear speed of a point on the rim in m/s.

Example 9

challenge
A sector has perimeter 2424 cm (two radii plus the arc). For what central angle θ\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°60° to radians, (b) 270°270° to radians, (c) π3\dfrac{\pi}{3} to degrees, (d) 7π6\dfrac{7\pi}{6} to degrees.

Example 2

medium
A clock's minute hand is 1515 cm long. How far does its tip travel in 2020 minutes? How large is the sector area swept?

Example 3

easy
Convert 180°180° to radians.

Example 4

easy
Convert 90°90° to radians.

Example 5

easy
Convert π6\frac{\pi}{6} radians to degrees.

Example 6

easy
Convert 360°360° to radians.

Example 7

easy
Convert π\pi radians to degrees.

Example 8

easy
Convert 45°45° to radians.

Example 9

easy
Convert 2π3\frac{2\pi}{3} radians to degrees.

Example 10

easy
How many radians are in one full revolution?

Example 11

medium
Find the arc length subtended by a central angle of π3\frac{\pi}{3} radians in a circle of radius 6.

Example 12

medium
A central angle of 40°40° subtends an arc in a circle of radius 9. Find the arc length.

Example 13

medium
Convert 135°135° to radians.

Example 14

medium
Find the area of a sector with central angle π4\frac{\pi}{4} radians and radius 4.

Example 15

medium
Convert 7π6\frac{7\pi}{6} radians to degrees and name its quadrant.

Example 16

medium
A wheel turns through 55 radians. Through how many degrees does it turn? (Round to the nearest degree.)

Example 17

medium
Convert 5π3\frac{5\pi}{3} radians to degrees.

Example 18

medium
Convert 300°300° to radians.

Example 19

medium
Find the arc length for a central angle of 120°120° in a circle of radius 3.

Example 20

challenge
A circle has radius 10. An arc on it has length 1515. Find the central angle in radians, then in degrees (round to nearest degree).

Example 21

challenge
Two pulleys of radii 3 and 7 are connected by a belt. If the small pulley turns through 14π3\frac{14\pi}{3} radians, how far (arc length) does the belt move, and through what angle does the large pulley turn?

Example 22

challenge
Express the angle π180\frac{\pi}{180} radian in degrees, and explain what this number represents.

Example 23

easy
Convert 120∘120^\circ to radians.

Example 24

easy
Convert 3π4\dfrac{3\pi}{4} radians to degrees.

Example 25

easy
Convert 5π4\dfrac{5\pi}{4} radians to degrees.

Example 26

medium
Convert 315∘315^\circ to radians.

Example 27

medium
Find the area of a sector with radius 66 and central angle π3\dfrac{\pi}{3}.

Example 28

medium
A wheel of radius 0.50.5 m rolls without slipping. Through how many radians does it turn after traveling 44 m?

Example 29

medium
A circle has radius 99. Find the arc length subtended by an angle of 80∘80^\circ.

Example 30

medium
Find the radius of a circle if a 22-rad central angle subtends an arc of length 99.

Example 31

hard
A circular track has radius 5050 m. A runner runs 200200 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 rr and the arc length equals the radius. What is the central angle in radians?

Example 33

hard
On a circle of radius 1212, two radii form a central angle of 50∘50^\circ. Find the exact area of the sector.
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Background Knowledge

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

unit circlepi