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The set of points in space whose distance to a fixed point (the **center**) is
a fixed positive number (the **radius**) is a **sphere**.
A circle of radius *r* and center (*x*,*y*,*z*) has equation

(*x*-*x*)+(*y*-*y*)+(*z*-*z*)=*r*,

or

*x*+*y*+*z*-2*x**x*-2*y**y*-2*z**z*+*x*+*y*+*z*-*r*=0.

Conversely, an equation of the form

*x*+*y*+*z*+2*dx*+2*ey*+2*fz*+*g*=0

defines a sphere if *d*+*e*+*f*>*g*; the center is
(-*d*, -*e*, -*f*)
and the radius is .

Four points not on the same plane determine a unique sphere.
If the points have coordinates (*x*,*y*,*z*),
(*x*,*y*,*z*), (*x*,*y*,*z*) and (*x*,*x*,*z*), the equation
of the sphere is

Given two points *P*=(*x*,*y*,*z*) and
*P*=(*x*,*y*,*z*),
there is a unique sphere whose diameter is *P**P*;
its equation is

(*x*-*x*)(*x*-*x*)+(*y*-*y*)(*y*-*y*)+(*z*-*z*)(*z*-*z*)=0.

The **area** of a sphere of radius *r* is
4*r*, and the **volume** is *r*.

The **area of a spherical polygon** (that is, of a polygon on the
sphere whose sides are arcs of great circles) is

where *r* is the radius of the sphere, *n* is the number of vertices,
and the are the internal angles of the polygons in radians.
In particular, the sum of the angles of a spherical triangle is always
greater than =180°, and the excess is proportional to the
area.

Let the radius be *r* (Figure 1, left).
The **area** of the curved region is 2*rh*=*p*. The
**volume** of the cap is *h*(3*r*-*h*)=*h*(3*a*+*h*).

**Figure 1:** Left: a spherical cap. Middle: a spherical zone (of two
bases). Right: a spherical segment.

Let the radius be *r* (Figure 1, middle).
The **area** of the curved region is 2*rh*. The
**volume** of the zone is *h*(3*a*+3*b*+*h*).

Let the radius be *r* (Figure 1, right). The **area** of
the curved region (lune) is 2*r*, the angle being measured in
radians. The **volume** of the segment is *r*.

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.