Derivation of circumference of ellipse

How can we generalize the formula for the circumference of a circle, 2(pi) r ? In this formula there's only one occurrence of r . Actually, though, there are two of them; they're being added. The circumference of a circle is (pi)r + (pi)r . So we might conjecture that the circumference of an ellipse can be found by replacing one r with an a and the other with a b to get pia + pib , or pi(a + b ). Can it?

To test the conjecture we would prefer to consider cases in which it's easy to calculate the circumference. But the only case in which that's true is the circle, and we know that the conjecture holds in that case.

Although we can't actually calculate the circumference in other cases without knowing the formula, we can think about the extreme case of a very narrow ellipse. Here the circumference is very close to that of the major axis. If the major axis has length 2a , then the circumference would be close to 4a. But if b is very close to 0, then the expression (pi) (a + b ) is very close to (pi)a, which is less than 4a. This reasoning is not conclusive, but it does indicate that our formula may not be completely accurate.

How can we find a better expression? Let's try calculus. Here we're trying to find the arc length of a curve. The circumference of an ellipse centered at the origin will be 4 times the length of the piece in the first quadrant. We can solve the rectangular coordinate equation for y to get

Then we can apply the standard formula for arc length to get the definite integral

.