Equation of a circle calculator

This circle equation calculator displays a standard form equation of a circle, a parametric form equation of a circle, and a general form equation of a circle given the center and radius of the circle. Formulas can be found below the calculator.

Center

Standard form equation of a circle

General form equation of a circle

Parametric form equation of a circle

Equation of a circle

An equation of a circle is an algebraic way to define all points that lie on the circumference of the circle. That is, if the point satisfies the equation of the circle, it lies on the circle's circumference. There are different forms of the equation of a circle:

• general form
• standard form
• parametric form
• polar form.

General Form Equation of a Circle

The general equation of a circle with the center at $(x_0, y_0)$ and radius $r$ is
$x^2+ax+y^2+by+c=0$,
where
$a=-2x_0\\b=-2y_0\\c=x^2_0+y^2_0-r^2$
With general form, it is difficult to reason about the circle's properties, namely the center and the radius. But it can easily be converted into standard form, which is much easier to understand.

Standard Form Equation of a Circle

The standard equation of a circle with the center at $(x_0, y_0)$ and radius $r$ is
$(x^2-x_0) + (y^2-y_0)=r^2$
You can convert general form to standard form using the technique known as Completing the square. From this circle equation, you can easily tell the coordinates of the center and the radius of the circle.

Parametric Form Equation of a Circle

The parametric equation of a circle with the center at $(x_0, y_0)$ and radius $r$ is
$x=r cos \theta + x_0\\y=r sin \theta + y_0$
This equation is called "parametric" because the angle theta is referred to as a "parameter". This is a variable which can take any value (but of course it should be the same in both equations). It is based on the definitions of sine and cosine in a right triangle.

Polar Form Equation of a Circle

The polar form looks somewhat similar to the standard form, but it requires the center of the circle to be in polar coordinates $(r_0, \phi)$ from the origin. In this case, the polar coordinates on a point on the circumference $(r, \theta)$ must satisfy the following equation
$r^2-2r r_0 cos(\theta - \phi)+r^2_0=a^2$,
where a is the radius of the circle.

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PLANETCALC, Equation of a circle calculator