Equation of a Circle
The standard form of the equation of a circle with centre (h, k) and radius r.
Formula
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Example
Centre (3, −2), radius 5 → (x − 3)² + (y + 2)² = 25
How it works
Every point on a circle is exactly the same distance — the radius — from the centre. The equation (x − h)² + (y − k)² = r² captures this directly: the left side is the squared distance from any point (x, y) to the centre (h, k), and setting it equal to r² means that distance is exactly r. If the centre is at the origin, h and k are both zero and the equation simplifies to x² + y² = r².
When to use it
Use this when you need to write, graph, or analyse a circle on a coordinate plane — for instance, finding which points lie on a circle, determining the centre and radius from an equation, or solving where a line intersects a circle.
Worked example, step by step
Write the equation of a circle with centre (3, −2) and radius 5. Step 1: Identify h = 3, k = −2, r = 5 Step 2: Substitute into (x − h)² + (y − k)² = r² (x − 3)² + (y − (−2))² = 5² (x − 3)² + (y + 2)² = 25 Final answer: (x − 3)² + (y + 2)² = 25
Common mistakes to avoid
- Forgetting to change the sign of k when substituting — if the centre is (3, −2), the equation has (y + 2), not (y − 2)
- Forgetting to square the radius on the right-hand side — writing r instead of r²
- Confusing the standard form with the general (expanded) form and losing track of h and k
Did You Know?
Expanding the standard form gives the general equation x² + y² + Dx + Ey + F = 0, where D = −2h, E = −2k, and F = h² + k² − r².
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