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

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 PP; 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 4r, 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 2rh=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 2rh. 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.

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.