3.2 Applications of Radian Measure

Arc Length on a Circle

Arc Length – the length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians is given by the product of the radius and the radian measure of the angle –

(Note: The formula for arc length if θ is in degrees is

- that is “a part of the circumference”

** θ MUST BE IN RADIANS ** if using

Example

A circle has a radius 25.60 cm. Find the length of the arc intercepted by a certain angle having each of the following measures.

(ans: 22.4π~70.37 cm, 7.68π~24.13 cm)

a)

b) 54^o

Find the radius of a circle which has a central angle of intersecting an arc of length
Example

Erie, Pa is approximately due north of Columbia, South Carolina. The latitude of Erie is 42^o N. The latitude of Columbia is 34^o N. Find the north-south distance between the two cities. (ans: 890 km)

Example

A rope is being wound around a drum with radius .327 ft. How much rope will be wound around the drum if the drum is rotated through and angle of 132.6^o? (ans: .757 ft)

Example

Two gears are adjusted so that the smaller gear drives the larger one of radius 5.4 in. If the smaller gear of radius 3.6 in. rotates through an angle of 150^o, through how many degrees will the larger gear rotate? (ans: 100^o)

Area of a Sector of a circle

A sector of a circle is the portion of the interior of a circle intercepted by a central angle.

Area of a Sector

The area A of a sector of a circle of radius r and central angle θ is given by

** AGAIN, θ MUST BE IN RADIANS **

Example

Find the area of a sector of a circle having radius 15.20 ft and central angle 108.0^o. (ans: 217.8 ft²)

Find the measure (in degrees) of the central angle of a sector of a circle if the area of the sector is sq. units and the radius is 3 units.