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What Is the Equation for a Sphere?

A sphere

You can think of a sphere as the outer layer of a basketball. The surface of a sphere can be described mathematically by this equation:

Theory

The Equation of a Sphere

The surface of a sphere with radius $r_{0}$ and its center at the point $(x_{0}, y_{0}, z_{0})$ is described by the equation

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2} = r_{0}^{2} .

Most of the time, you will not get the finished version of this equation in your tasks. Instead, you have to rewrite it by “completing the squares”, using that

x^{2} + b x + {(\frac{b}{2})}^{2} = {(x + (\frac{b}{2}))}^{2} .

You can follow these steps to create the squares:

Rule

Determine if an Expression is an Equation for a Sphere

You start with a quadratic expression on this form:

x^{2} + b x + y^{2} + d y + z^{2} + f z = g .

(1)

Note! The coefficients for the quadratic terms are 1.

1.

You first work with the

x

terms:

x^{2} + b x .

Add

{(\frac{b}{2})}^{2}

on each side of Equation (1). This gives you

x^{2} + b x + {(\frac{b}{2})}^{2} + \dots = {(\frac{b}{2})}^{2} + \dots

2.

You can now factorize the

x

terms on the left side:

x^{2} + b x + {(\frac{b}{2})}^{2} = {(x + (\frac{b}{2}))}^{2}

Repeat these steps with the $y$ terms and then the $z$ terms.

3.

You can now express the whole left-hand side of the equation as a sum of squares. You place all the constants on the right-hand side and add them together. If the right-hand side is positive, it can be written as

r^{2}

and the expression is the equation for a sphere:

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2} = r^{2} .

These steps may seem daunting, but might be easier to understand through an example.

Example 1

This might be the equation for a sphere:

x^{2} + 4 x + y^{2} - 2 y + z^{2} = 4 .

Find out if it is, and if so, determine the center and radius.

To determine this, you need to use the steps from above. The $x$ , $y$ and $z$ terms needs to be rewritten as complete squares.

Items 1 and 2.

These steps can be done at the same time, as the goal is to complete the squares. To make $x^{2} + 4 x$ a square, you need to divide 4 (the number in front of $x$ ) by 2, and add the square of this to both sides of the equation. You get:

\begin{array}{l} x^{2} + 4 x + {(\frac{4}{2})}^{2} + \dots & = \dots + {(\frac{4}{2})}^{2} \\ {(x + \frac{4}{2})}^{2} + \dots & = \dots + {(\frac{4}{2})}^{2} \end{array}

To make $y^{2} - 2 y$ a square, add ${(\frac{- 2}{2})}^{2}$ on both sides. You get:

\begin{array}{l} y^{2} - 2 y + {(\frac{- 2}{2})}^{2} + \dots & = \dots + {(\frac{- 2}{2})}^{2} \\ {(y + \frac{- 2}{2})}^{2} + \dots & = \dots + {(\frac{- 2}{2})}^{2} \end{array}

You see that $z^{2}$ already is a complete square, so you can leave it alone.

Item 3.

Now, all you have to do is combine the calculations above:

\begin{array}{l} x^{2} + 4 x + y^{2} - 2 y + z^{2} & = 4 \\ {(x + \frac{4}{2})}^{2} + {(y + \frac{- 2}{2})}^{2} + z^{2} & = 4 + {(\frac{4}{2})}^{2} \\ + {(\frac{- 2}{2})}^{2} \\ {(x + 2)}^{2} + {(y - 1)}^{2} + z^{2} & = 4 + 4 + 1 \\ {(x + 2)}^{2} + {(y - 1)}^{2} + z^{2} & = 3^{2} \end{array}

\begin{array}{l} x^{2} + 4 x + y^{2} - 2 y + z^{2} & = 4 \\ {(x + \frac{4}{2})}^{2} + {(y + \frac{- 2}{2})}^{2} + z^{2} & = 4 + {(\frac{4}{2})}^{2} + {(\frac{- 2}{2})}^{2} \\ {(x + 2)}^{2} + {(y - 1)}^{2} + z^{2} & = 4 + 4 + 1 \\ {(x + 2)}^{2} + {(y - 1)}^{2} + z^{2} & = 3^{2} \end{array}

This is the equation of a sphere and the radius is

r = 3

and

s = (- 2, 1, 0)

. is the center.

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