Frustum of a cone: surface and volume

Text

A frustum of a cone has base radii $8$ cm and $3$ cm and slant height $13$ cm. Find the lateral surface, the total surface, the height and the volume.

Solution

Lateral surface: $S_\ell=\pi(R+r)\,a=\pi\cdot11\cdot13=143\pi\approx449.25$ cm$^2$. Total surface: $S_t=143\pi+\pi(64+9)=143\pi+73\pi=216\pi\approx678.58$ cm$^2$. Height: $h=\sqrt{a^2-(R-r)^2}=\sqrt{169-25}=12$ cm. Volume: $V=\dfrac{\pi h}{3}(R^2+r^2+Rr)=\dfrac{\pi\cdot12}{3}(64+9+24)=4\pi\cdot97=388\pi\approx1218.94$ cm$^3$. $\boxed{S_\ell=143\pi;\ S_t=216\pi\ \text{cm}^2;\ h=12\ \text{cm};\ V=388\pi\approx1218.94\ \text{cm}^3}$