Spherical Segment of One Base

Spherical Segment of One Base

Each of the two portions of a sphere obtained by cutting the sphere with a plane.

The outer layer of the spherical segment of one base is called a spherical cap.


The lateral area A of a spherical cap of radius r and height h is given by: A = \(2πrh \).

The volume V of a spherical segment is obtained by using the formula : V = \(\dfrac{πh(3q^{2} + h^{2})}{6}\) where \(h\) is the height of the segment and \(q\) is the radius of the small circle.

The Pythagorean theorem can be applied to the right triangle shown in the figure above, in which \(\left ( R-h \right )^{2}+r^{2}=R^{2}\), to deduce that \(r^{2}=2Rh-h^{2}\), where the radius \(r\) of the spherical segment has a value of \(\sqrt{h(2R-h)}\).The preceding formula to calculate the volume of a spherical segment as a function of the height \(h\) and the radius \(R\) of the sphere becomes: \(V = \dfrac{\pi h^2 (3r-h)}{3}\).


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