Circle equation is one of 6 circles formulas in the geometry section of this library, and it is used at high school level.
Why circle equation works
A point belongs to the circle exactly when its distance to the centre equals the radius. Writing that distance out with the Pythagorean relation and then squaring both sides to clear the root produces this equation, so it is a distance condition with the root removed.
What each symbol means
$(h,k)$ is the center and $r$ is radius.
Circle equation: when it holds
$r\ge0$; both coordinate axes must use the same scale.
When it stops applying
It describes a real circle only when the right-hand side is positive and the two squared terms carry matching coefficients. After completing the square, a right-hand side of 0 leaves a single point and a negative one leaves no points at all.
Circle equation: a worked example
Center $(2,-1)$ and radius $3$ give $(x-2)^2+(y+1)^2=9$.
The mistake to avoid
What people do: Reading the centre straight off the signs that appear in the equation.
Why it goes wrong: The form subtracts the centre's coordinates, so a plus sign inside the brackets already means a negative coordinate, and the centre gets reported in the wrong quadrant.
Do this instead: Rewrite any plus as a subtraction of a negative number, then read the coordinate off after the minus sign.
Circle equation: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $(h,k)$ is the center and $r$ is radius.
- Check the conditions before substituting. $r\ge0$; both coordinate axes must use the same scale.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Circles
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where circle equation comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Equation of a Circle — the lesson behind this formula: read and build circle equations on the coordinate plane.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about circle equation
How do I find the centre and radius from an expanded equation?
Complete the square in each variable. Doing that to the equation x² + y² − 4x + 6y = 12 turns it into a circle of radius 5 centred at (2,−3).
Why is the radius squared on the right-hand side?
Because both sides were squared to remove the square root from the distance condition, so a right-hand side of 9 corresponds to a radius of 3.
What changes when the centre is at the origin?
Both centre coordinates are 0, so the brackets disappear and the equation shortens to the sum of the two squared coordinates equalling the radius squared.
Is this equation a function?
No. Almost every horizontal position corresponds to two heights, one above the centre and one below, so it fails the vertical line test.