Equation of a Circle

The equation of a circle can be derived using Pythagoras Theorem. The circle below has centre

\[(x_1,y_1)\]

. The point

\[(x,y)\]

lies on the circle.

Construct a right angled triangle with hypotenuse from the centre

\[(x_1,y_2)\]

to the point

\[(x,y)\]

- and so equal to the radius

\[r\]

of the circle, base parallel to the

\[x\]

- axis and height parallel to the

\[y\]

- axis.
Then the base of the triangle is

\[x-x_1\]

and the height is

\[y-y_1\]

.
From Pythagoras Theorem,

\[(x-x_1)^2+(y-y_1)^2=r^2\]