Spherical Cap — from Wolfram MathWorld

But why is a sphere’s surface area four times its shadow?
But why is a sphere’s surface area four times its shadow?

A spherical cap is the region of a sphere which lies above (or below) a given plane. If the plane passes through the center of the sphere, the cap is a called a hemisphere, and if the cap is cut by a second plane, the spherical frustum is called a spherical segment. However, Harris and Stocker (1998) use the term “spherical segment” as a synonym for what is here called a spherical cap and “zone” for spherical segment.

Let the sphere have radius , then the volume
of a spherical cap of height and base radius is given by the equation of a spherical
segment

(1)

with ,
giving

(2)

Using the Pythagorean theorem gives

(3)

which can be solved for
as

(4)

so the radius of the base circle is

(5)

and plugging this in gives the equivalent formula

(6)

In terms of the so-called contact angle (the angle between the normal to the sphere at the bottom of the cap and the base plane)

(7)

(8)

so

(9)

The geometric centroid occurs at a distance

(10)

above the center of the sphere (Harris and Stocker 1998, p. 107).

The cap height
at which the spherical cap has volume equal to half a
hemisphere is given by

(11)

Consider a cylindrical box enclosing the cap so that the top of the box is tangent to the top of the sphere. Then the enclosing box has volume

(12)

(13)

(14)

so the hollow volume between the cap and box is given by

(15)

The surface area of the spherical cap is given by the same equation as for a general zone:

(16)

(17)

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