Problem

Find the circumference of the hypocycloid

Solution

This particular kind of hypocycloid is called an astroid, because of its starlike shape. Obviously, this will depend on the value of a. Here's the graph of this shape for a = 4.

For any choice of a, the curve will extend to the points (–a, 0) and (a, 0), and from (0, –a) to (0, a).

This curve is symmetric with respect to the x and y axes. To find its length, we'll find the length of the curve in the first quadrant, from x = 0 to x = a, and multiply by 4.

We need dy/dx to set up the integral. Solve for y as a function of x.

We want the square of this for the integral.

The length of the curve in the first quadrant is

This integral has a neat solution.

The total length of the curve is