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8. Applications of Radian Measure

by M. Bourne

In this section, we see some of the common applications of radian measure, including arc length, area of a sector of a circle, and angular velocity.

Go back to the section on Radians if you are not sure what is going on.

Arc Length

Flash Interactive

Don't miss the interactive Flash game in this section.

Go to
Angular velocity game

Arc with length s

The length, s, of an arc of a circle radius r subtended by θ (in radians) is given by:

s = r θ

If r is in meters, s will also be in meters. Likewise, if r is in cm, s will also be in cm.

Example 1

Find the length of the arc of a circle with radius `4\ "cm"` and central angle `5.1` radians.

Area of a Sector

The area of a sector with central angle θ (in radians) is given by:

Area of a sector of a circle

`"Area"=(theta\ r^2)/2`

If r is in `"m"`, the area will be measured in `"m"`2. If r is in `"cm"`, area will be in `"cm"`2.

Example 2

Find the area of the sector with radius `7\ "cm"` and central angle `2.5` radians.

Angular Velocity

The time rate of change of angle θ by a rotating body is the angular velocity, written ω (omega). It is measured in radians/second.

If v is the linear velocity (in m/s) and r is the radius of the circle (in m), then

v =

Note: If r is in `"cm"`, v will be in `"cm/s"`.

Example 3

A bicycle with tyres `90\ "cm"` in diameter is travelling at `25` km/h. What is the angular velocity of the tyre in radians per second?

Flash Game

The man runs at a constant velocity and a ball is revolving overhead.

You increase points if:

  • the linear velocity of the ball is more than 6 m/sec - add 1 point
  • the ball is close to the man (within one body length) - add 10 points

Your points go down if:

  • the ball goes outside the borders - minus 1 point
  • the ball hurts the man - minus 20 points

You can use

  • the first slider or the right and left arrows on your keyboard to change the radius and
  • the second slider or the up and down arrows to change the angular velocity.
Loading Flash movie...


1. A section of side walk is a circular sector of radius `1.25\ "m"` and central angle `50.6°`. What is the area of this section of sidewalk?

2. A cam is in the shape of a circular sector with radius `1.875\ "cm"` and central angle `165.58°`. What is the perimeter of the cam?

3. The roller on a computer printer makes `2200` rev/min. What is its angular velocity?

4. The propeller on a motorboat is rotating at `130` rad/s. What is the linear velocity of a point on the tip of a blade if the blade is `22.5` cm long?

5. The sweep second hand of a watch is `15.0` mm long. What is the linear velocity of the tip?

Pulley Problems


You can investigate the linear velocity of a belt moving around two pulleys in this Flash example.

Go to Pulleys simulation.

Online Algebra Solver

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Trigonometry Lessons on DVD


Easy to understand trigonometry lessons on DVD. See samples before you commit.

More info: Trigonometry videos

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