# 14 Spheres

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 -2xx -2yy -2zz +x  +y  +z  -r =0.

Conversely, an equation of the form

x +y +z +2dx+2ey+2fz+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.

### Spherical cap

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 (3r-h)=  h(3a +h ). Figure 1: Left: a spherical cap. Middle: a spherical zone (of two bases). Right: a spherical segment.

### Spherical zone (of two bases)

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

### Spherical segment and lune

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

Next: 15 Surfaces of Revolution. The Torus
Up: Part II: Three-Dimensional Geometry
Previous: 13.3 Cones The Geometry Center Home Page

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.