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Coordinate geometry

Equation of a circle

Every point (x, y) on a circle with centre (h, k) and radius r satisfies this equation.

(x−h)2+(y−k)2=r2
Centre at the originx2+y2=r2
General formx2+y2+Dx+Ey+F=0

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

−12−10−8−6−4−224681012−6−4−2246(h, k)r
  • Centre (2, −1), radius 5
  • Every point (x, y) on the circle is r away from the centre, which is the distance formula squared.
  • Expanded: x² + y² − 4x + 2y = 20

What the symbols mean

r
the radius
h
the x-coordinate of the centre
k
the y-coordinate of the centre
x
the x-coordinate of a point on the circle
y
the y-coordinate of a point on the circle

When to use it

Use it to write the equation of a circle from its centre and radius, to read the centre and radius off an equation, to find the radius from the centre and one point, to test whether a point is inside (left side less than r2), on, or outside the circle, and to find the points of the circle at a given x or y (two, one or none).

Worked example

A circle has centre (2, −1) and passes through (5, 3). Find its radius.

  1. Start from the formula.

    (x−h)2+(y−k)2=r2

  2. Rearrange to make r the subject: take the square root of both sides (r is positive).

    r=(x−h)2+(y−k)2

  3. Substitute h = 2, k = −1, x = 5, y = 3 and work it out.

    r=(5−2)2+(3−(−1))2
    =32+42
    =9+16
    =25
    =5

Answer r = 5

Why it works

A circle is all the points at distance r from the centre. Write that with the distance formula, (x−h)2+(y−k)2=r, and square both sides.

Common mistakes

  • Getting the centre's signs backwards: (x+3)2+(y−4)2=25 has centre (−3, 4).
  • Reading r2 as the radius: with 25 on the right the radius is 5.
  • Expanding (x−h)2 as x2−h2. It is x2−2hx+h2.