Find center and radius of circle given two points calculator

Table of Contents Show

Equation of a circle
General Form Equation of a Circle
Standard Form Equation of a Circle
Parametric Form Equation of a Circle
Polar Form Equation of a Circle
How to find a circle passing through 3 given points

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

and radius

, where

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 and radius is

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 and radius is

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

from the origin. In this case, the polar coordinates on a point on the circumference

must satisfy the following equation

,
where a is the radius of the circle.

Calculation precision

Digits after the decimal point: 2

Equation of a circle in standard form

Equation of a circle in general form

Parametric equations of a circle

How to find a circle passing through 3 given points

Let's recall how the equation of a circle looks like in general form:

Since all three points should belong to one circle, we can write a system of equations.

The values

and

are known. Let's rearrange with respect to unknowns a, b and c.

Now we have three linear equations for three unknowns - system of linear equations with the following matrix form:

We can solve it using, for example, Gaussian elimination like in Gaussian elimination. No solution means that points are co-linear, and it is impossible to draw a circle through them. The coordinates of a center of a circle and it's radius related to the solution like this

Knowing center and radius, we can get the equations using Equation of a circle calculator

This calculator will find either the equation of the circle from the given parameters or the center, radius, diameter, circumference (perimeter), area, eccentricity, linear eccentricity, x-intercepts, y-intercepts, domain, and range of the entered circle. Also, it will graph the circle. Steps are available.

Related calculators: Parabola Calculator, Ellipse Calculator, Hyperbola Calculator, Conic Section Calculator

Your Input

Find the center, radius, diameter, circumference, area, eccentricity, linear eccentricity, x-intercepts, y-intercepts, domain, and range of the circle $$$x^{2} + y^{2} = 9$$$.

Solution

The standard form of the equation of a circle is $$$\left(x - h\right)^{2} + \left(y - k\right)^{2} = r^{2}$$$, where $$$\left(h, k\right)$$$ is the center of the circle and $$$r$$$ is the radius.

Our circle in this form is $$$\left(x - 0\right)^{2} + \left(y - 0\right)^{2} = 3^{2}$$$.

Thus, $$$h = 0$$$, $$$k = 0$$$, $$$r = 3$$$.

The standard form is $$$x^{2} + y^{2} = 9$$$.

The general form can be found by moving everything to the left side and expanding (if needed): $$$x^{2} + y^{2} - 9 = 0$$$.

Center: $$$\left(0, 0\right)$$$.

Radius: $$$r = 3$$$.

Diameter: $$$d = 2 r = 6$$$.

Circumference: $$$C = 2 \pi r = 6 \pi$$$.

Area: $$$A = \pi r^{2} = 9 \pi$$$.

Both eccentricity and linear eccentricity of a circle equal $$$0$$$.

The x-intercepts can be found by setting $$$y = 0$$$ in the equation and solving for $$$x$$$ (for steps, see intercepts calculator).

x-intercepts: $$$\left(-3, 0\right)$$$, $$$\left(3, 0\right)$$$