Find the circumference of the circle x² + y² = a²

Find the circumference of the circle $x^2 + y^2 = a^2$.

We parametrize the circle and apply the arc-length formula directly.

Step 1: Parametrize

\[x=a\cos t,\qquad y=a\sin t,\qquad t\in[0,2\pi]\]

Step 2: Arc length formula

\[L = \int_0^{2\pi}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\]

Step 3: Compute the integrand

$\frac{dx}{dt}=-a\sin t$, $\frac{dy}{dt}=a\cos t$, so:

\[\sqrt{a^2\sin^2t+a^2\cos^2t} = \sqrt{a^2} = a\]

Step 4: Integrate

\[L = \int_0^{2\pi}a\,dt = 2\pi a\]

\[\boxed{L = 2\pi a}\]

Step 5: Cross-check in polar form

With $\rho=a$ constant, $\frac{d\rho}{d\varphi}=0$, so $L=\int_0^{2\pi}\sqrt{a^2+0}\,d\varphi = 2\pi a$. ✓

Summary

\[\boxed{L = 2\pi a}\]

Key insight: The Pythagorean identity $\sin^2t+\cos^2t=1$ makes the circle's arc-length integrand exactly constant, which is precisely why $\pi$ can be defined as the ratio of circumference to diameter.