Position of a point relative to a circle

Whether a point is inside a circle, on it, or outside is found by substituting the point's coordinates into the circle's equation and comparing. Take the circle $x^2 + y^2 = 25$ (center at the origin, radius $5$).

The square of the distance from a point $(x, y)$ to the center $(0, 0)$ is $x^2 + y^2$. Compare this with the square of the radius, $25$.

• $x^2 + y^2 < 25$: nearer than the radius → inside the circle
• $x^2 + y^2 = 25$: exactly the radius → on the circle
• $x^2 + y^2 > 25$: farther than the radius → outside the circle

Check with three points. $(1, 2)$ gives $1^2 + 2^2 = 5 < 25$, so it is inside; $(4, 3)$ gives $4^2 + 3^2 = 25$, so it is on the circle; $(4, 4)$ gives $4^2 + 4^2 = 32 > 25$, so it is outside.

So comparing the sum of squares $x^2 + y^2$ with $25$ tells you the position without ever taking a square root. The large dots on the graph are the inside point $(1, 2)$, the on-circle point $(4, 3)$, and the outside point $(4, 4)$.