Cone - volume, surface area and net

Statement

A cone has radius $3$ and height $4$. Calculate its volume, the total surface area and the net angle.

Solution

Slant height: $\ell = \sqrt{r^2+h^2} = \sqrt{9+16} = 5$. Volume: $V = \tfrac{1}{3}\pi r^2 h = \tfrac{1}{3}\pi\cdot 9\cdot 4 = \boxed{12\pi}.$ Total surface area: $S_{\text{tot}} = \pi r(r+\ell) = \pi\cdot 3\cdot 8 = \boxed{24\pi}.$ Net angle: $\beta = \frac{360°\cdot r}{\ell} = \frac{360°\cdot 3}{5} = \boxed{216°}.$

Links

Topics: Synthetic geometry in space
Concepts: Apothem · Cone · Circular sector · Surface area · Volume
Skills: Calculating · Using formulae
Exercise types: Geometric problem