# How do you find the parametric equations of a circle?

##### 1 Answer

We'll start with the parametric equations for a circle:

#y = rsin t#

#x = rcos t#

where

If you know that the implicit equation for a circle in Cartesian coordinates is

We will take the equation for

#x/r = cos t#

#t = arccos (x/r)#

Now substitute into the equation for

#y = rsin arccos(x/r)#

Thus,

#y = r*sqrt(r^2 - x^2)/r#

This simplifies to

#y = sqrt(r^2 - x^2)#

If we square this entire deal and solve for

#r^2 = x^2 + y^2#

which is precisely the equation for a circle in Cartesian coordinates.