How to find the center, radius, and equation of a sphere

A sphere is made up of points that are all the same distance from its center. That distance gives us a way to build the equation, and the equation gives us a way to read the center and radius. Let’s work in both directions.

Use the distance from the center

Let the center be (h,k,l) and let (x,y,z) be a point on the sphere. The distance formula gives the radius. Squaring both sides gives the standard equation.

(x-h)2+(y-k)2+(z-l)2=r2
Radius from the center of a sphere to its surfaceA line segment joins the center to a point on the sphere. Its length is r, the radius.CenterrSurface point
Every point on the sphere is the same distance from the center.

Write the equation from the center and a point

Suppose the center is (1,1,2) and the sphere passes through (2,4,6). The changes in the three coordinates are 1, 3, and 4. Add their squares to find r2.

r2=12+32+42=26r=26
(x-1)2+(y-1)2+(z-2)2=26

If you expand the squares and collect constants, the same equation becomes

x2-2x+y2-2y+z2-4z=20

Find the center and radius by completing the square

x2+2x+y2-2y+z2-6z=14

Group each squared term with its linear term. Take half of each linear coefficient, then square it. Those three values tell us what to add to both sides.

x2+2x+1=(x+1)2y2-2y+1=(y-1)2z2-6z+9=(z-3)2

The right side becomes 14+1+1+9=25. Write the equation with the three completed squares.

(x+1)2+(y-1)2+(z-3)2=25

In the first factor, x+1 is x-(-1), so the first coordinate of the center is -1. The other two coordinates are 1 and 3. The center is (-1,1,3) and the radius is 25=5.

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