8.1 Circle Equations

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Main Content

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What is a Circle?

A circle is the set of all points in a flat plane that are the exact same distance from one fixed point. That fixed point is called the centre, and that fixed distance is called the radius.

Think of it like this: if you tie a piece of string to a pin on a piece of paper, hold the string tight, and trace all the way around — every point your pencil touches is the same distance from the pin. That traced path is a circle.

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Deriving the Equation of a Circle

To write an equation that describes a circle, we use Pythagoras' theorem — the rule that says in a right-angled triangle, the square of the longest side equals the sum of the squares of the other two sides.

Here is how we do it:

• Let the centre of the circle be the point $C(a, b)$
• Let any point on the circle be $P(x, y)$
• The distance from $C$ to $P$ is always equal to the radius $r$

If you draw a right-angled triangle between $C$ and $P$:

• The horizontal side has length $x - a$
• The vertical side has length $y - b$
• The hypotenuse (the slanted side from centre to point) has length $r$

Applying Pythagoras' theorem:

$\boxed{(x - a)^2 + (y - b)^2 = r^2}$

This is the equation of a circle with centre $(a, b)$ and radius $r$.

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Form 1 — Completed Square Form

$\mathbf{(x - a)^2 + (y - b)^2 = r^2}$

This is the most useful form because the centre and radius can be read off directly.