# Equation of Unit Circle

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## 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.

$\blacksquare$

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 3$. Roots of Unity - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**polar equation**