Find the arc length of the cardioid rho = a(1 - cos phi)

Find the arc length of the cardioid $\rho = a(1 - \cos\varphi)$.

We apply the polar arc-length formula and simplify using the half-angle identity.

Step 1: Arc length formula in polar coordinates

\[L = \int_0^{2\pi}\sqrt{\rho^2+\left(\frac{d\rho}{d\varphi}\right)^2}\,d\varphi, \qquad \frac{d\rho}{d\varphi}=a\sin\varphi\]

Step 2: Simplify the integrand

\[\rho^2+\left(\frac{d\rho}{d\varphi}\right)^2 = a^2(1-\cos\varphi)^2+a^2\sin^2\varphi = a^2\left[1-2\cos\varphi+1\right] = 2a^2(1-\cos\varphi)\]

Step 3: Use the half-angle identity

Since $1-\cos\varphi=2\sin^2\frac\varphi2$, and $\sin\frac\varphi2\ge0$ for $\varphi\in[0,2\pi]$:

\[\sqrt{2a^2(1-\cos\varphi)} = 2a\sin\frac\varphi2\]

Step 4: Integrate

Let $u=\frac\varphi2$: $L=\int_0^{2\pi}2a\sin\frac\varphi2\,d\varphi = 4a\int_0^\pi\sin u\,du = 4a[-\cos u]_0^\pi = 4a(1-(-1)) = 8a$.

\[\boxed{L = 8a}\]

Step 5: Geometric check

The cardioid's maximum radius is $\rho_{\max}=2a$ (at $\varphi=\pi$); its perimeter $8a$ is exactly four times this maximum radius.

Summary

\[\boxed{L = 8a}\]

Key insight: Despite the cardioid's cusp at the origin, its total perimeter is exactly $8a$ — four times its maximum radius — one of the cleanest results in the classical study of polar curves.