Equations of Circles

A circle is every point a fixed distance from a fixed point. Write that with the distance formula and you have its equation.

A garden sprinkler at the point (6, 4) throws water five metres in every direction.

Which points get wet? Not the ones nearer than 5 — those get wet too. Which points are exactly on the edge?

a sprinkler that reaches five metres 5 m (6, 4) every wet point is exactly 5 m from the sprinkler — and no dry one is

A point is on the edge exactly when its distance from (6, 4) is 5.

You already know how to say that: √(x-6)2 + (y-4)2 = 5.

Square both sides and the root disappears: (x-6)2 + (y-4)2 = 25.

That is the whole equation of a circle, and it is not a new idea — it is the distance formula with the distance fixed. Centre (h, k), radius r: (x-h)2 + (y-k)2 = r2.

a sprinkler that reaches five metres 5 m (6, 4) every wet point is exactly 5 m from the sprinkler — and no dry one is

Reading the equation

In (x-h)2 + (y-k)2 = r2 the centre is (h, k) and the radius is r.

The signs flip: (x - 6)2 means h = 6, but (x + 3)2 means h = -3, because x + 3 is x - (-3).

And the number on the right is r2, not r. If it says 25, the radius is 5.

The circle equation as Pythagoras: from the centre (h, k) to a point (x, y) on the circle, the legs are x − h and y − k and the hypotenuse is the radius r every point (x, y) on the circle is r from the centre (h, k) — that is Pythagoras xy hk x − h y − k r (h, k)(x, y) (x − h)² + (y − k)² = r² leg² + leg² = hypotenuse² the signs flip (x − 6)² means h = 6 (x + 3)² means h = −3 because x + 3 = x − (−3) the right side is r² = 25 means the radius is 5

Two sign traps in one line

(x+2)2 + (y-5)2 = 49 has centre (-2, 5) and radius 7. The x sign flips, the y sign flips, and the 49 must be rooted — three chances to slip in one equation.

Writing one from scratch

Given a centre and a radius, substitute straight in. Centre (-1, 4), radius 6: (x+1)2 + (y-4)2 = 36.

Given a centre and a point on the circle, find the radius with the distance formula first.

Given the two ends of a diameter, the centre is the midpoint and the radius is half the distance between them.

From general form back to standard

x2 + y2 - 6x + 8y - 11 = 0 hides its centre. To find it, complete the square in x and again in y.

Group: (x2 - 6x) + (y2 + 8y) = 11. Half of -6 is -3, squared is 9. Half of 8 is 4, squared is 16.

Add both to both sides: (x-3)2 + (y+4)2 = 36. Centre (3, -4), radius 6.

Add to both sides

Completing the square adds something to the left, so the same amount must go on the right — and you do it twice, once for x and once for y.

Is a point inside, on, or outside?

Put the point into the left-hand side and compare with r2.

Less than r2 means inside; equal means on; greater means outside.

You are just comparing the squared distance with the squared radius, so no roots are needed.

Testing points against the circle x² + y² = 25: (−2, 1) gives 5, inside; (3, 4) gives 25, on; (4, −4) gives 32, outside substitute the point and compare with r² — no square roots needed xy 5 (−2, 1) (3, 4) (4, −4) x² + y² = 25, so r² = 25 (−2, 1): 4 + 1 = 55 < 25 → inside (3, 4): 9 + 16 = 2525 = 25 → on (4, −4): 16 + 16 = 3232 > 25 → outside

Circles and lines together

To find where a line meets a circle, substitute the line into the circle equation and solve the resulting quadratic.

Two solutions means the line cuts the circle; one means it is a tangent; none means it misses.

The discriminant tells you which before you finish solving — the same b2 - 4ac from algebra, doing geometry.

Worked examples

Reading off centre and radius

Give the centre and radius of (x+2)2 + (y-5)2 = 49.

  1. (x + 2) is (x - (-2)), so h = -2. The sign flips.
  2. (y - 5) gives k = 5 directly. No flip needed here.
  3. r2 = 49, so r = 7. Root the right-hand side.
  4. Centre (-2, 5), radius 7. All three traps handled.

Centre (-2, 5), radius 7

From a diameter

A circle has a diameter from (1, 2) to (7, 10). Write its equation.

  1. Centre is the midpoint: (1+72, 2+102) = (4, 6) Midpoint adds and halves.
  2. Diameter length: √62 + 82 = 10 Distance between the endpoints.
  3. Radius is half: r = 5 Half the diameter.
  4. (x-4)2 + (y-6)2 = 25 Substitute centre and r2.

(x-4)2 + (y-6)2 = 25

Completing the square

Find the centre and radius of x2 + y2 - 6x + 8y - 11 = 0.

  1. Group and move the constant: (x2 - 6x) + (y2 + 8y) = 11 Constants to the right.
  2. Half of -6 is -3; (-3)2 = 9. Half of 8 is 4; 42 = 16. Two completions.
  3. Add both to both sides: (x-3)2 + (y+4)2 = 11 + 9 + 16 = 36 Balance is everything here.
  4. Centre (3, -4), radius 6. Check: 36 = 62 ✓

Centre (3, -4), radius 6

Words to know

Standard form
(x-h)2 + (y-k)2 = r2, centre (h,k) and radius r.
General form
x2 + y2 + Dx + Ey + F = 0, which hides the centre until you complete the square.
Radius
The fixed distance. Note the equation stores r2, not r.
Completing the square
The move that turns general form back into standard form.

Mistakes to watch for

Reading the radius straight off the right-hand side. The equation stores r2, so = 25 means a radius of 5. Forgetting to root it makes every subsequent answer five times too big.

Getting the sign of the centre wrong. (x + 3)2 means h = -3, because the standard form subtracts. Both coordinates can flip independently, so check each one.

Adding to only one side when completing the square. You add 9 to the left, so 9 must go on the right — and you do it twice, once for each variable. Forgetting shrinks the radius.

Halving the diameter's endpoints instead of finding the midpoint. The centre is the average of the two points, not half of one of them — and the radius is half the distance, not half a coordinate.

The equation shows why π never appears in it. π is about a circle's length and area; the equation is only about which points belong.

Descartes and Fermat independently invented this way of writing shapes as equations in the 1630s.

It made a whole class of problems mechanical that had needed ingenuity for two thousand years — you can now find where a line meets a circle by solving, not by drawing.

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