# What is the equation of a sphere in standard form?

Feb 1, 2015

The answer is: ${x}^{2} + {y}^{2} + {z}^{2} + a x + b y + c z + d = 0$,

This is because the sphere is the locus of all
points $P \left(x , y , z\right)$ in the space whose distance from $C \left({x}_{c} , {y}_{c} , {z}_{c}\right)$ is equal to r.

So we can use the formula of distance from $P$ to $C$, that says:

$\sqrt{{\left(x - {x}_{c}\right)}^{2} + {\left(y - {y}_{c}\right)}^{2} + {\left(z - {z}_{c}\right)}^{2}} = r$ and so:

${\left(x - {x}_{c}\right)}^{2} + {\left(y - {y}_{c}\right)}^{2} + {\left(z - {z}_{c}\right)}^{2} = {r}^{2}$,

${x}^{2} + 2 \left(x\right) \left({x}_{c}\right) + {x}_{c}^{2} + {y}^{2} + 2 \left(y\right) \left({y}_{c}\right) + {y}_{c}^{2} + {z}^{2} + 2 \left(z\right) \left({z}_{c}\right) + {z}_{c}^{2} = {r}^{2}$,

${x}^{2} + {y}^{2} + {z}^{2} + a x + b y + c z + d = 0$,

in which

$a = 2 {x}_{c}$;
$b = 2 {y}_{c}$;
$c = 2 {z}_{c}$;
$d = {x}_{c}^{2} + {y}_{c}^{2} + {z}_{c}^{2} - {r}^{2}$;

So:

$C \left(- \frac{a}{2} , - \frac{b}{2} , - \frac{c}{2}\right)$

and $r$, if it exists, is:

$r = \sqrt{{x}_{c}^{2} + {y}_{c}^{2} + {z}_{c}^{2} - d}$.

If the center is in the Origin, than the equation is:

${x}^{2} + {y}^{2} + {z}^{2} = {r}^{2}$,