Planar development of the cone

By “unrolling” the lateral surface of the cone we obtain a notable plane figure: a circular sector.

Property — Planar development of the cone

If the cone is “cut” along the slant height and “unrolled”, we obtain a circular sector of radius $\ell$ and arc $2\pi r$. The central angle of this sector is $\beta = \frac{2\pi r}{\ell} \text{ rad} = \frac{360°\cdot r}{\ell}.$

On the left the cone with radius $r$, height $h$ and slant height $\ell$; on the right its planar development, a circular sector of radius $\ell$ and central angle $\beta$.

Links

Topics: Synthetic geometry of space
Concepts: Cone · Circular sector · Planar development
Skills: Using formulas