How Many Sides?
Divide the circumference of a circle by its radius. For any circle, the result is \[ \pi \approx 3.1415926535 \dots \] The digits of \(\pi\) don't follow a known pattern. So how do we know the 10th digit? Or the 10000th?
The perimeter of a regular polygon is an easy-to-compute approximation of a circle's circumference. You can find \(pi\) to any precision by adding sides to the polygon.
Suppose you wanted a more precise approximation of \(\pi\), e.g., more than 4 digits after the decimal. One approach is to approximate the circumference by the perimeter of a regular polygon. As you increase the number of sides, the polygon gets closer and closer to a circle.
How many sides does the polygon need to have to obtain the known 5th digit of \(\pi\)?