# 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

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

s=rθ

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 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=rω

**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**.

### Exercises:

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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