The Equation of a Circle
A circle is every point that sits the same distance — the radius
r — from a fixed centre. If the centre is the origin, then for any
point (x, y) on the circle the
distance from
the origin is exactly r. The distance formula is just Pythagoras, so
x^2 + y^2 = r^2.
That single equation captures the whole circle: a point lies on it precisely when its
coordinates satisfy x^2 + y^2 = r^2.
Where the equation comes from
Step through the picture: drop a point on the circle down to the
x-axis and a right triangle appears, with legs
x and y and hypotenuse
r. Pythagoras finishes the job.
Moving the centre
If the centre is not the origin but a point (a, b), the same
argument measures distance from there instead. Replacing x with
x - a and y with
y - b gives the general circle:
(x - a)^2 + (y - b)^2 = r^2,
a circle of radius r centred at (a, b).
- centred at the origin:
x^2 + y^2 = r^2;
- centred at (a, b):
(x - a)^2 + (y - b)^2 = r^2;
- the radius is r = \sqrt{r^2} — take the square
root of the right-hand side, not the number itself;
- the signs flip: a centre of (3, -2) appears as
(x - 3)^2 + (y + 2)^2.