Is a Point Inside or Outside a Circle?

To decide whether a point lies inside, on, or outside a circle, compare its distance from the centre with the radius. It’s a short, reliable calculation that also underpins whether a line meets a circle.

The rule

For a point $P$ and a circle with centre $C$ and radius $r$, compute the distance $CP$:

• $CP < r$: $P$ is inside the circle.
• $CP = r$: $P$ is on the circle.
• $CP > r$: $P$ is outside the circle.

To avoid surds, it’s often cleaner to compare $CP^2$ with $r^2$ directly.

A worked example

Determine whether the point $P(5, 5)$ lies inside, on, or outside the circle $x^2 + y^2 = 25$. Centre $C(0, 0)$, radius $r = 5$, so $r^2 = 25$. $CP^2 = 5^2 + 5^2 = 25 + 25 = 50$. Since $CP^2 = 50 > 25 = r^2$, the point is outside the circle.

Comparing the squares ($50$ vs $25$) avoids working with $\sqrt{50}$, quicker and less error-prone.

Reading the centre and radius first

If the circle is given in expanded form (like $x^2 + y^2 - 6x + 4y - 12 = 0$), complete the square first to read the centre and radius, then apply the distance test.

Common mistakes

• Comparing $CP$ with $r^2$ (mixing a distance with a squared radius), compare like with like.
• Forgetting to find the centre when the equation is in expanded form.
• Reversing the inequality (inside vs outside).

Full topic context: Circle Geometry notes, and the tangent-length worked example.