8. Applications of Radian Measure

by M. Bourne

Linear velocity applet

Don't miss the interactive angular/linear velocity animation on this page.

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Angular velocity applet

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

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

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

Interactive linear velocity applet


A car is going around a circular track of radius 0.5 km. The speedometer in the car shows the (magnitude) of the linear velocity.

At first, the car goes around the track once in just over 6 minutes. It's angular velocity is 1 rad/min or one complete revolution in 2π = 6.28 min.

The distance travelled in this time is the circumference of the circle, C = 2πr = 2π(0.5) = 3.14 km. So the car is travelling at `(3.14" km")/(6.28 min) = 0.5" km/min" = 30" km/h"`.

The linear velocity showing on the speedo is 30 km/h.

Things to do

In this applet, you can:

  • Vary the radius of the track
  • Vary the angular velocity of the car

Observe the change in linear speed as you do so.

Of course, angle measures are in radians in this applet.

r = 0 km ω = 0 rad/min

v = = 0 × 0 = 0 km/min = 0 km/h


Angular velocity:

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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.


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