To find the length of an entire circle, inscribe a regular polygon in it and increase the number of sides. What do you notice? To find the length of an arc of the circle, find the arc measure to know how much of the circle it represents and multiply it by the length of the entire circle.
See solution.
Practice makes perfect
Let's consider the circle C with radius r shown below.
We are interested in finding the circumference of the circle above. Since we know how to find the length of straight line segments, we will inscriberegular polygons in the circle C, then increase the number of sides.
Notice that as the number of sides increases, the polygon approaches the circle. By finding the perimeter of the polygon we will approximate the circumference of the circle. Let P be the circumference of the circle.
Next, let's suppose that we want to find the length of just one portion of the circle — the length of an arc.
To find the length of AB we start by finding the measure of arc AB, for which we find the measure of ∠ ACB. Once we know it, we divide it by 360^(∘), and that way we will know how much of the circle is represented by the arc.
Portion of the Circle = mAB/360^(∘)
Finally, the length of AB is found by multiplying the expression above by the length of the entire circle.
Length of AB = mAB/360^(∘)* P
Keep in mind that we did not write an explicit formula to find the circumference of a circle. That formula will be developed in this chapter.
