Exercise 52 Page 804

We are asked to justify the formula for the length of an arc. To do so, let's recall that an arc length s equals the product of the radius r and central angle θ (in radians). s= r * θ Now, we will remember the circumference of a circle. Notice that this is also the length of an arc that measures 2π radians.

Then, we will divide the whole circumference by 2π to calculate the length of an arc which measures 1 radian. 2π r/2π = r Notice that the length of an arc which measures 1 radian is the same as the length of the radius of the circle.

Since the arc length of each radian is one unit of radius, we can find the arc length of any central angle θ (in radians) by multiplying the central angle θ by the radius r. ccc Angle & & Arc Measure 1 radian & ⇒ & r * 1 θ radians & ⇒ & r * θ Therefore, we can easily use the formula s= r * θ to find the length of an arc.