Circle Equation and Graph Calculator

Quick Examples

How to Use This Calculator

This online circle equation calculator is an analytical tool that allows you to process different geometric data to obtain the exact formulas and all the properties of the figure. The mathematical engine interprets the input information and returns a detailed report along with the graphical representation.

To get started, use the main dropdown menu and indicate to the system what information you have. The available calculation modes are the following:

1. Center and radius: Enter the Cartesian coordinates of the center (h, k) and the length of the radius. With these direct parameters, the solver will instantly construct the corresponding equations.
2. Equation: The system is capable of interpreting the circle's equation in any of its forms. You can input the general equation (x² + y² + Dx + Ey + F = 0), the standard form, or unsimplified and unarranged expressions.
3. Center and a point: Input the coordinates of the center and those of any point (x, y) that belongs to the circle. The solver will calculate the distance between them to deduce the radius.
4. Diameter endpoints: Provide the coordinates of two opposite points that form a diameter. The calculator will locate the center at the midpoint of the segment and obtain the length of the radius.
5. Three points: Enter the coordinates of three non-collinear points through which the curve passes. The tool will solve the necessary system to find the unique circle that contains them.

Once your data is processed, you will get the results organized into boxes, accompanied by their visual representation:

• Equations box: This includes the standard form (ideal for quickly visualizing the center and radius) and the general form expanded and set equal to zero.
• Elements box: A summary with all the metric and geometric properties of the figure. Here you will find the coordinates of the center, the lengths of the radius and diameter, the values for the perimeter (circumference) and area, as well as the exact points of intersection with the Cartesian x and y axes.
• Interactive graph: An interactive Cartesian plane will be generated at the bottom. You will be able to observe the plotting of the circle, with its center marked and the equation visible. The canvas allows you to zoom in, zoom out, or pan the view to analyze the curve.

All input fields accept integers, decimals, and exact fractions. To ensure mathematical precision, irrational magnitudes like π or square roots will be kept intact in their symbolic form in the results, also accompanied by their respective decimal approximation.

Solved Exercises

The following are examples of problems solved by the calculator.

Calculate the equation and elements of the circle with its center at the origin (0, 0) and radius r = 6.

Equations of the circle

Standard form equation

$$ x^2 + y^2 = 36 $$

General form equation

$$ x^2 + y^2 - 36 = 0 $$

Main elements

Center: \( C \left(0, 0\right) \)

Radius: \( r = 6 \)

Diameter: \( D = 12 \)

Circumference: \( P = 12\pi \approx 37.70 \)

Area: \( A = 36\pi \approx 113.1 \)

x-axis intercepts

$$ x_1 = -6 \\[1em] x_2 = 6 $$

y-axis intercepts

$$ y_1 = -6 \\[1em] y_2 = 6 $$

Determine the standard form equation and the elements of the circle whose general form equation is \( x^2+y^2+6x-8y-11=0. \)

Equations

Standard form equation

$$ \left(x + 3\right)^2 + \left(y - 4\right)^2 = 36 $$

General form equation

$$ x^2 + y^2 + 6x - 8y - 11 = 0 $$

Main elements

Center: \( C \left(-3, 4\right) \)

Radius: \( r = 6 \)

Diameter: \( D = 12 \)

Circumference: \( P = 12\pi \approx 37.70 \)

Area: \( A = 36\pi \approx 113.1 \)

x-axis intercepts

$$ x_1 = \dfrac{-4 \sqrt{5}-6}{2} \approx -7.47 \\[1em] x_2 = \dfrac{4 \sqrt{5}-6}{2} \approx 1.47 $$

y-axis intercepts

$$ y_1 = \dfrac{-6 \sqrt{3}+8}{2} \approx -1.20 \\[1em] y_2 = \dfrac{6 \sqrt{3}+8}{2} \approx 9.20 $$

Find the general and standard form equations of the circle with center at C(-2, 3) that passes through the point (1, -1).

Equations of the circle

Standard form equation

$$ \left(x + 2\right)^2 + \left(y - 3\right)^2 = 25 $$

General form equation

$$ x^2 + y^2 + 4x - 6y - 12 = 0 $$

Main elements

Center: \( C \left(-2, 3\right) \)

Radius: \( r = 5 \)

Diameter: \( D = 10 \)

Circumference: \( P = 10\pi \approx 31.42 \)

Area: \( A = 25\pi \approx 78.54 \)

x-axis intercepts

$$ x_1 = -6 \\[1em] x_2 = 2 $$

y-axis intercepts

$$ y_1 = \dfrac{-2 \sqrt{21}+6}{2} \approx -1.58 \\[1em] y_2 = \dfrac{2 \sqrt{21}+6}{2} \approx 7.58 $$

Find the equations, center, radius, and other elements of the circle knowing that the points (-4, 2) and (6, 4) are endpoints of a diameter.

Equations

Standard form equation

$$ \left(x - 1\right)^2 + \left(y - 1\right)^2 = 34 $$

General form equation

$$ x^2 + y^2 - 2x - 2y - 32 = 0 $$

Main elements

Center: \( C \left(1, 1\right) \)

Radius: \( r = \sqrt{34} \approx 5.83 \)

Diameter: \( D = 2 \sqrt{34} \approx 11.66 \)

Circumference: \( P = 2 \sqrt{34}\pi \approx 36.64 \)

Area: \( A = 34\pi \approx 106.81 \)

x-axis intercepts

$$ x_1 = \dfrac{-2 \sqrt{33}+2}{2} \approx -4.74 \\[1em] x_2 = \dfrac{2 \sqrt{33}+2}{2} \approx 6.74 $$

y-axis intercepts

$$ y_1 = \dfrac{-2 \sqrt{33}+2}{2} \approx -4.74 \\[1em] y_2 = \dfrac{2 \sqrt{33}+2}{2} \approx 6.74 $$

Calculate the equations and elements of the circle that passes through the points (0, 0), (0, 3), and (4, 0).

Equations of the circle

Standard form equation

$$ \left(x - 2\right)^2 + \left(y - \dfrac{3}{2}\right)^2 = \dfrac{25}{4} $$

General form equation

$$ x^2 + y^2 - 4x - 3y = 0 $$

Main elements

Center: \( C \left(2, \dfrac{3}{2}\right) \approx \left(2, 1.5\right) \)

Radius: \( r = \dfrac{5}{2} = 2.5 \)

Diameter: \( D = 5 \)

Circumference: \( P = 5\pi \approx 15.71 \)

Area: \( A = \dfrac{25}{4}\pi \approx 19.63 \)

x-axis intercepts

$$ x_1 = 0 \\[1em] x_2 = 4 $$

y-axis intercepts

$$ y_1 = 0 \\[1em] y_2 = 3 $$

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