# Equation of Sphere/Rectangular Coordinates/Corollary

## Theorem

The equation of a sphere with radius $R$ whose center is at the origin expressed in Cartesian coordinates is:

$x^2 + y^2 + z^2 = R^2$

## Proof

From Equation of Sphere in Rectangular Coordinates, the equation of a sphere with radius $R$ and center $\tuple {a, b, c}$ expressed in Cartesian coordinates is:

$\paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$

Setting $a = b = c = 0$ yields the result.

$\blacksquare$