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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

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:

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

 

Answer

s = rθ

= 4 × 5.1

= 20.4 cm

Area of a Sector

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

If r is in m, the area will be measured in m². If r is in cm, area will be in cm².

Example:

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

 

Answer

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:

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?

 

Answer

Now v = r ω

The units are a mix of cm and km. Let's present everything in meters.

We need to convert v to m/s first.

25 km/h = 25000 m/h = 25000/3600 m/s = 6.94444 m/s

Also, we have r = (90 cm)/2 = 45 cm = 0.45 m

So ω = v/r = 6.94444/0.45 = 15.43 rad/s

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?

Answer

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?

Answer

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

Answer

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?

Answer

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

Answer

Pulley Problems

Go to Pulleys simulation.

Need a break? Play a math game. Well, they all involve math... No, really!

Chapter Contents

• Trigonometric Functions
• 1. Angles
• 2. Sine, Cosine, Tangent & Reciprocals
• 3. Values of Trigonometric Functions
• 4. The Right Triangle and Applications
• 5. Signs of the Trigonometric Functions
• 6. Trigonometric Functions of Any Angle
• 7. Radians
• 8. Applications of Radian Measure
• Pulleys
• 9. Radians and the Trigonometric Ratios

• Comments, Questions?

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