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