Spherical zones and caps

Notes by R. Bigoni
(from Note Didattiche)


1. Spherical zone

Given a spherical surface Σ with center O and radius OP with measure R, two parallel planes φ and ψ such that their distance from the center O is smaller than R, intersect Σ in two circumferences.
The region of Σ bounded by these circumferences is called spherical zone Ζ. The circles enclosed by these circumferences are its bases.


To calculate the surface of a spherical zone we can proceed in the following way:

2. Spherical segment

With reference to fig. 1, the solid Ω bounded by the spherical zone Z and its bases is called spherical segment.

With reference to fig. 3 and with a method similar to that used for the calculation of S, the volume V of Ω can be calculated as the sum of the volumes of the cylinders with infinitesimal height dy and base radius Eqn007.gif with y from yA to yB;






From the equation (4) we get


Moreover, if we apply the Pythagorean theorem


and subtract side by side



If we plug this expression for yA into the equation (12)



3. Spherical cap

If in fig. 1 the plane most distant from the center is tangent to Σ in P, the resulting figure is said spherical cap.


In fig. 4, H is the center of the base of the cap, P is its vertex, the segment HK is the radius of the base with measure r, the segment HP is the height of the cap with measure δ and OK is the radius of Σ with measure R.

The equation (5) allows the calculation of the outer surface of the cap


Th equation (14) with xA=r and xB=0 gives the volume of the cap


If we apply the Pythagorean theorem, we get


and, plugging this expression for r2 into the equation (15), we obtain



3. Visible cap of a spherical surface.

Given a spherical surface Σ with center O and radius R and the point E outside Σ at distance h from it, the portion of Σ observable from E is a cap whose area can be calculated as follows:

The surface of Σ is S=4πR2, hence the fraction of Σ visible from E is


last updated: March 16, 2018