# Equation of Unit Circle

## Theorem

Let the unit circle have its center at the origin of the Cartesian $xy$ coordinate plane.

Its equation is given by:

$x^2 + y^2 = 1$

## Proof

From Equation of Circle, the equation of a circle with radius $R$ and center $\tuple {a, b}$ is:

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

Substituting $\tuple {0, 0}$ for $\tuple {a, b}$ and $1$ for $R$ gives the result.

$\blacksquare$