Circle Equation Derivation: Distance Formula & Pythagorean Theorem

The equation of a circle can be deduced using the distance formula and the Pythagorean theorem.

Consider a circle with center (h,k) and radius r. Let (x,y) be any point on the circle.

The distance between (x,y) and (h,k) is given by the distance formula:

d = sqrt((x-h)^2 + (y-k)^2)

Since (x,y) is on the circle, its distance from the center is r:

d = r

Substituting the value of d from the first equation into the second equation:

sqrt((x-h)^2 + (y-k)^2) = r

Squaring both sides:

(x-h)^2 + (y-k)^2 = r^2

This is the equation of a circle with center (h,k) and radius r.