Arc Length Exercise

See if you can figure out how to solve the following problem before you read the solution, just by using your newly acquired knowledge of how arc length, angle when measured in radians, and radius are related along with your previous knowledge of area, circumference, and proportions. Recall that the area of an entire circle is πr² and the circumference of an entire circle is 2πr.

EXAMPLE Suppose the arc length of a circular sector is 4.5cm and the central angle is 156°.  What is the area of the circular sector?

Solution: We need to find the radius and to do that we need the angle to be in radians. That way we will be able to use the relastionship: θ = s/r. Thus, first let's convert 156° to radians:

Hence the angle θ = 13π/15, so now we can find the radius: .
Now that we have r, we can calculate the area of our circular wedge. We have 156/360 of a circle so our area is: .