Equilateral cylinder inscribed in a sphere

Text

An equilateral cylinder (height equal to the base diameter) is inscribed in a sphere of radius $10$ cm. Find the radius and height of the cylinder and its total surface area.

Solution

For the equilateral cylinder $h=2r$; the axial cross-section is a square of side $2r$, whose diagonal is the sphere’s diameter: $2r\sqrt2=2R\ \Rightarrow\ r=\frac{R}{\sqrt2}=\frac{10}{\sqrt2}=5\sqrt2\approx7.07\ \text{cm},\quad h=2r=10\sqrt2\approx14.14\ \text{cm}.$ Total surface of the equilateral cylinder: $S_t=2\pi r(r+h)=2\pi r\cdot3r=6\pi r^2=6\pi\cdot50=300\pi\approx942.48$ cm$^2$. $\boxed{r=5\sqrt2\approx7.07;\ h=10\sqrt2\approx14.14\ \text{cm};\ S_t=300\pi\approx942.48\ \text{cm}^2}$