The equation of a circle

Tie a goat to a post with a 5 m rope and let it wander to the very end in every direction. The path it traces is a circle: every point exactly 5 m from the post.

That's the definition. A circle is the set of all points at a fixed distance $r$ (the radius) from a fixed point (the centre).

Where the goat analogy breaks: real goats wander inside the circle too. The circle is only the edge.

Take the circle with centre $C(2, 1)$ and radius 5. Let $P(x, y)$ be any point on it.

Draw a right-angled triangle with $CP$ as the long side (the hypotenuse):

• across: from $x = 2$ to $x$, a distance of $x - 2$
• up: from $y = 1$ to $y$, a distance of $y - 1$
• hypotenuse: $CP = 5$, the radius

Pythagoras says (across)² + (up)² = (hypotenuse)²:

$(x - 2)^2 + (y - 1)^2 = 5^2$

If $P$ is to the left of or below $C$, these come out negative. Squaring makes them positive, so the equation still works.

This is true for every point on the circle, and for no other point. So it is the equation of the circle.

Two traps, both about signs and squares:

• The centre's coordinates have the opposite sign to the numbers in the brackets: $(x + 3)^2 + (y - 1)^2 = 16$ has centre $(-3, 1)$.
• The right-hand side is $r^2$, so take the square root: radius $\sqrt{16} = 4$, not 16.

Think of $(x + 3)$ as $(x - (-3))$, and the sign flip stops being mysterious.