Find equation of circle with center and tangent line calculator

Equation of a circle given the centre and the radius

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

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.

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

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

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

The domain is $$$\left[h - r, h + r\right] = \left[-3, 3\right]$$$.

The range is $$$\left[k - r, k + r\right] = \left[-3, 3\right]$$$.

Answer

Standard form: $$$x^{2} + y^{2} = 9$$$A.

General form: $$$x^{2} + y^{2} - 9 = 0$$$A.

Graph: see the graphing calculator.

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

Radius: $$$3$$$A.

Diameter: $$$6$$$A.

Circumference: $$$6 \pi\approx 18.849555921538759$$$A.

Area: $$$9 \pi\approx 28.274333882308139$$$A.

Eccentricity: $$$0$$$A.

Linear eccentricity: $$$0$$$A.

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

y-intercepts: $$$\left(0, -3\right)$$$, $$$\left(0, 3\right)$$$A.

Domain: $$$\left[-3, 3\right]$$$A.

Range: $$$\left[-3, 3\right]$$$A.

How do you find the equation of a circle with two points and tangent?

How do you find the equation of a circle with only the center?