The Equation of a Circle
A radar screen sweeps a ring around the airport, a lawn sprinkler wets a circular patch of
grass, and a phone mast reaches everyone within a fixed range. Each of these is a circle — a set
of points all the same distance from a centre — and to get a computer to work with one, we need
a way to write that circle down as an equation.
You already know how to pin a straight line to an equation:
y = mx + c. Every point on the line obeys it, and no other point
does. A circle is just as tidy — it has its own equation, and once you can read it,
a circle stops being a picture and becomes something you can calculate with.
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. The equation is secretly
Pythagoras' theorem, applied to every point on the circle at once.
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:
x^2 + y^2 = r^2.
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).
Read it out loud as a promise: "the horizontal gap x - a and the
vertical gap y - b, squared and added, always come to
r^2."
- 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.
Worked examples
1 · Write the equation. A circle has centre the origin and radius
6. The right-hand side is the radius squared, so
x^2 + y^2 = 6^2 = 36.
Not x^2 + y^2 = 6 — that would be a circle of radius
\sqrt{6} \approx 2.45.
2 · Read off centre and radius. Given
(x - 3)^2 + (y + 2)^2 = 25, match it against
(x - a)^2 + (y - b)^2 = r^2. The x-part
is (x - 3), so a = 3. The
y-part is (y + 2) = (y - (-2)), so
b = -2. And r^2 = 25, so
r = \sqrt{25} = 5.
\text{centre } (3, -2), \quad \text{radius } 5.
3 · On, inside, or outside? Where does
(6, 8) sit relative to the circle
x^2 + y^2 = 25? Compute the left-hand side for that point and
compare it with r^2:
6^2 + 8^2 = 36 + 64 = 100 \; > \; 25.
Because 100 is bigger than 25,
the point is farther than the radius from the centre — it lies outside. The
rule: compare x^2 + y^2 with r^2 —
less means inside, equal means on the circle, greater means
outside.
-
The centre's signs flip. In
(x - a)^2 + (y - b)^2 = r^2 the centre is
(a, b) with minus signs built in. So
(x - 3)^2 + (y + 2)^2 has centre
(3, -2) — the +2 means
b = -2, because (y + 2) = (y - (-2)).
Reading it as (3, 2) is the single most common mistake here.
-
The right-hand side is r^2, not
r. The circle
x^2 + y^2 = 25 has radius 5, not
25 — always take the square root. And going the other way, a
radius of 7 gives x^2 + y^2 = 49, not
7.
The equation is Pythagoras wearing a disguise: it literally says the horizontal and vertical
distances from the centre, squared and added, equal the radius squared. That "distance equals a
constant" idea refuses to stay in two dimensions. Add a third axis and
x^2 + y^2 + z^2 = r^2 is a sphere; keep going and
you get the abstract "balls" of higher-dimensional geometry, all defined by the same one line.
It's also doing quiet work in the real world. Every time a game decides whether you're inside a
blast radius, or a map app shows "shops within 2 km", the computer isn't drawing a circle —
it's checking whether x^2 + y^2 is less than
r^2. And GPS pins you down by
trilateration: each satellite knows you sit on a sphere of some radius around it, and
where those spheres cross is you. The humble circle equation is the maths behind knowing where
anything is.
See it explained