Spherical cap and section circle

Text

A sphere of radius $10$ cm is cut by a plane at distance $6$ cm from the center. Find the area of the smaller cap and the radius of the section circle.

Solution

The height of the smaller cap is $h=R-d=10-6=4$ cm. Cap area: $\mathcal A=2\pi R h=2\pi\cdot10\cdot4=80\pi\approx251.33$ cm$^2$. Section-circle radius: $r=\sqrt{R^2-d^2}=\sqrt{100-36}=8$ cm. $\boxed{\mathcal A_\text{cap}=80\pi\approx251.33\ \text{cm}^2;\ r_\text{sec}=8\ \text{cm}}$