The equation of a circle

A circle is the set of all points that are the same distance from one fixed point¹. That fixed point is the centre, and the shared distance is the radius. Here we take a circle with centre at the origin and radius $5$ .

Building the equation

Pick any point $(x, y)$ on the circle. Measured from the centre it is $x$ across and $y$ up, which makes a right triangle, so by the Pythagorean theorem the two legs squared add up to the radius squared.

x^2 + y^2 = 5^2 = 25

Testing points

Point	$x^2 + y^2$	Position
$(3, 4)$	$25$	on the circle
$(5, 0)$	$25$	on the circle
$(-4, 3)$	$25$	on the circle
$(1, 1)$	$2$	inside

You can test whether a point is on the circle just by checking whether it satisfies the equation.

The general equation

The idea is the same for any radius. If the centre moves to $(a, b)$ , the horizontal gap becomes $x - a$ and the vertical gap $y - b$ .

(x - a)^2 + (y - b)^2 = r^2

Centre	Equation
$(0, 0)$	$x^2 + y^2 = r^2$
$(a, b)$	$(x-a)^2 + (y-b)^2 = r^2$

Why the graph is drawn in two pieces

Solving $x^2 + y^2 = 25$ for $y$ gives $y = \sqrt{25 - x^2}$ for the top half and $y = -\sqrt{25 - x^2}$ for the bottom half, and putting them together makes the whole circle. The two halves join at the ends $(\pm 5, 0)$ , where $y = 0$ .

Because one value of $x$ has two matching values of $y$ , a circle is not a function of the form $y = f(x)$ , which is why it is drawn as two separate curves.

The large dots on the graph are the centre and the point $(3, 4)$ on the circle. No matter which direction you go from the centre, the distance out to the circle is always the radius, and that is exactly what the equation says.

Circle, Wikipedia