All subjects · Mathematics · WMA12 (P2)

The equation of a circle

Every point on a circle is the same distance from the centre. Pythagoras turns that fact into an equation.

Centre and radius form

A circle with centre (a, b) and radius r has equation

(x - a)2 + (y - b)2 = r2.

Watch the signs: (x + 4)2 means a = -4.

Expanded form

Equations like x2 + y2 + 2fx + 2gy + c = 0 are circles too. Complete the square in x and in y to find the centre and radius.

Inside, on or outside

Substitute a point into (x - a)2 + (y - b)2. Less than r2: inside. Equal: on the circle. Greater: outside.

Worked example

Find the centre and radius of x2 + y2 - 6x + 4y - 12 = 0.

  1. (x - 3)2 - 9 + (y + 2)2 - 4 - 12 = 0
  2. (x - 3)2 + (y + 2)2 = 25

Answer: Centre (3, -2), radius 5

Key idea: (x - a)2 + (y - b)2 = r2 has centre (a, b) and radius r. Complete the square to convert the expanded form.

Check yourself

  1. What is the centre of (x + 4)2 + (y - 1)2 = 9?
    • (4, -1)
    • (4, 1)
    • (-4, 1)
    • (-4, -1)
    Show answer

    (-4, 1). (x - a)2 with a = -4 gives (x + 4)2. So the centre is (-4, 1).

  2. What is the radius of (x + 4)2 + (y - 1)2 = 9?
    Show answer

    3. r2 = 9, so r = 3.

  3. Find the radius of x2 + y2 + 8x - 2y - 8 = 0.
    Show answer

    5. (x + 4)2 - 16 + (y - 1)2 - 1 - 8 = 0, so (x + 4)2 + (y - 1)2 = 25 and r = 5.

  4. x2 + y2 - 10x + 6y + k = 0 has radius 6. Find k.
    Show answer

    -2. (x - 5)2 + (y + 3)2 = 25 + 9 - k. Set 34 - k = 36, so k = -2.

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

    5. r = sqrt(32 + 42) = 5.

  6. Is the point (1, 2) inside, on or outside the circle (x - 3)2 + (y - 2)2 = 9?
    • On the circle
    • Inside
    • Outside
    • It cannot be decided
    Show answer

    Inside. (1 - 3)2 + (2 - 2)2 = 4, which is less than 9.

  7. Which is the equation of the circle with centre (0, 5) and radius 4?
    • x2 + (y + 5)2 = 16
    • x2 + (y - 5)2 = 4
    • (x - 5)2 + y2 = 16
    • x2 + (y - 5)2 = 16
    Show answer

    x2 + (y - 5)2 = 16. a = 0, b = 5 and r2 = 16.

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Last updated 7 November 2026. Spotted a mistake? Tell us.