Loading calculator…

What is a Circle Equation Calculator?

A Circle Equation Calculator is a digital math tool designed to determine the equation of a circle in various forms. A circle is a 2D shape consisting of a set of points equidistant from a single center point. In mathematics, a circle equation can be expressed in three main forms: general form (x² + y² + Ax + By + C = 0), standard form (x - h)² + (y - k)² = r², and equations passing through specific points. This calculator is essential for high school students (grades 10-12) studying circle equations in mathematics. Circle equation material is an important part of analytic geometry, which also serves as a foundation for other geometry topics.

Forms of Circle Equations

(x-h)² + (y-k)² = r² or x² + y² + Ax + By + C = 0Formula: Center (h,k), Radius r, Coefficients A, B, C

Variables:

  • (x,y)Point Coordinates
    A point on the Cartesian plane(e.g.: (3, 4))
    💡 Determining position on the circle
  • (h,k)Center Point
    The center of the circle on the coordinate plane(e.g.: (2, -3))
    💡 Determining the middle location of the circle
  • rRadius
    Distance from center to the edge of the circle(e.g.: r = 5)
    💡 Determining the size of the circle
  • A, B, CGeneral Form Coefficients
    Constants in the equation x²+y²+Ax+By+C=0(e.g.: A=-4, B=6, C=-12)
    💡 Solving the general form

Steps to Determine a Circle Equation

To find the equation of a circle, first determine what information is known: center and radius (go directly to standard form), coefficients (general form), or points on the circle (substitution).

  1. 1Determine the known information (center & r, coefficients, or points)
  2. 2Select the appropriate sub-calculator (standard/general/points)
  3. 3Enter the known values into the form
  4. 4The calculator will generate the equation in complete form

Categories:

Standard Form(x-h)²+(y-k)²=r²
General Formx²+y²+Ax+By+C=0
Through 3 PointsSubstitute into x²+y²+Ax+By+C=0
ConversionComplete the square: general → standard

How to Use the KalkuLab Circle Equation Calculator

Enter center coordinates and radius to find the circle equation and properties:

  1. 1

    Select Input Mode

    Choose standard form (x-h)² + (y-k)² = r² or general form x² + y² + Dx + Ey + F = 0.

  2. 2

    Enter Known Values

    Enter center (h, k) and radius r, or coefficients D, E, F for general form.

  3. 3

    Calculate

    Press Calculate to get the equation, center, radius, and area.

  4. 4

    View Results

    See standard and general forms, center coordinates, radius, circumference, and area.

💡 Tip:

  • •Standard form: (x-h)² + (y-k)² = r², center (h,k), radius r
  • •General form: x² + y² + Dx + Ey + F = 0
  • •Center from general form: (-D/2, -E/2)
  • •Radius: r = √(h² + k² - F) from general form
  • •Area = πr², Circumference = 2πr

Examples

Example 1: Circle from Center and Radius

Problem:

Find the equation of a circle with center (3, -2) and radius 4.

Solution:
  1. 1.(x - 3)² + (y - (-2))² = 4²
  2. 2.(x - 3)² + (y + 2)² = 16
Result:(x - 3)² + (y + 2)² = 16

Center (3, -2), radius 4, area ≈ 50.27 square units.

Example 2: Find Center from General Form

Problem:

Find center and radius of x² + y² - 6x + 4y - 12 = 0

Solution:
  1. 1.Complete square: (x-3)² + (y+2)² = 25
  2. 2.Center: (3, -2), r = 5
Result:Center (3, -2), r = 5

The circle has center (3, -2) and radius 5.

Frequently Asked Questions

What is the equation of a circle?
Standard form: (x-h)² + (y-k)² = r² where (h,k) is the center and r is the radius. All points on the circle are distance r from the center.
How do you convert general form to standard form?
Complete the square for x and y terms. Example: x² + y² - 4x + 6y - 12 = 0 becomes (x-2)² + (y+3)² = 25, center (2,-3), radius 5.
What if r² is negative in general form?
If h² + k² - F < 0, the equation has no real circle (imaginary radius). The calculator will indicate this.
Is the KalkuLab Circle Equation Calculator free?
Yes, free to use on KalkuLab anytime.

Related Calculators

Quadratic Equation Calculator

Solve ax² + bx + c = 0

Area & Perimeter Calculator

Circle area and circumference

Pythagorean Calculator

Distance between points

References