According to Table 7.2.1, the arc length is obtained by evaluating the integral ${\int }_{{\mathrm{\theta }}_1}^{{\mathrm{\theta }}_2}\sqrt{r^2\+{\left(r\prime \right)}^2}\mathit{\ⅆ}\mathrm{\theta }$ for $r\=1\/2- \cos \left(\mathrm{\θ}\right)$. The radical in the integrand is easily seen to be

so the complete integration reduces to

The antiderivative of $\sqrt{5-4 \cos \left(\mathrm{\θ}\right)}$ is not an elementary function; it is a member of the family of elliptic functions that Maple knows. Without Maple, this integral could only be evaluated numerically, or with a table of integrals.

An elegant interactive solution is possible if the limaçon is defined as a function $R\left(\mathrm{\θ}\right)$.