Equation of a Circle Calculator - Circle Equation Calculator
Free equation of a circle calculator & circle equation calculator. Find circle equations in standard form (x - h)² + (y - k)² = r² and general form x² + y² + Dx + Ey + F = 0. Our calculator uses coordinate geometry principles to calculate center, radius, diameter, circumference, and area from given parameters.
Last updated: October 28, 2025
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Enter the radius of the circle (must be positive)
Circle Equation
Standard Form:
x² + y² = 25.00
General Form:
x² y² - 25 = 0
Center:
(0, 0)
Radius:
5.00
Diameter
10.00
Circumference
31.42
Area
78.54
Circle Equation Forms:
- • Standard: (x - h)² + (y - k)² = r² where (h, k) is center
- • General: x² + y² + Dx + Ey + F = 0 (expanded form)
- • Radius must be positive (r > 0)
- • Center at origin: x² + y² = r²
- • Area = πr², Circumference = 2πr
Circle Equation Calculator Types & Forms
Formula
(x - h)² + (y - k)² = r²
Shows center (h, k) and radius r clearly in the equation
Expanded form
x² + y² + Dx + Ey + F = 0
Expanded algebraic form with all terms on one side
Extract values
(h, k) and r
Identify center coordinates and radius from equation
Simplified form
x² + y² = r²
Simplest equation when center is at the origin
Formulas
C = 2πr, A = πr²
Calculate all circle properties from radius or diameter
Input
3 points
Determine unique circle passing through three non-collinear points
Quick Example Result
Circle with center (0, 0) and radius 5:
Standard Form
x² + y² = 25
General Form
x² + y² - 25 = 0
How Our Circle Equation Calculator Works
Our equation of a circle calculator uses coordinate geometry principles to generate circle equations in both standard and general forms. The calculation is based on the definition that a circle is the set of all points equidistant from a center point, expressed algebraically using the distance formula.
Circle Equation Formulas
The standard form clearly shows the center and radius, while the general form is useful for algebraic manipulation and solving systems of equations. Both forms represent the same circle.
Shows circle with labeled center, radius, and coordinate points
Mathematical Foundation
The circle equation derives from the distance formula. For a point (x, y) to be on a circle with center (h, k) and radius r, the distance from (x, y) to (h, k) must equal r. Using the distance formula: √((x-h)² + (y-k)²) = r. Squaring both sides gives the standard form: (x-h)² + (y-k)² = r². Expanding this algebraically yields the general form.
- Circle is defined as all points at distance r from center (h, k)
- Standard form directly shows center and radius
- General form results from expanding and rearranging standard form
- Completing the square converts general form back to standard form
- Circumference = 2πr, Area = πr², Diameter = 2r
- Circle at origin has equation x² + y² = r² (h = 0, k = 0)
Sources & References
- Geometry - Jurgensen, Brown, Jurgensen (Classic Edition)Standard reference for circle equations and coordinate geometry
- Precalculus - Stewart, Redlin, WatsonComprehensive coverage of conic sections and circle equations
- Wolfram MathWorld - CircleDetailed mathematical properties and equations of circles
Need help with other geometry tools? Check out our ellipse calculator and parabola calculator.
Get Custom Calculator for Your PlatformCircle Equation Calculator Examples
Given Information:
- Center: (h, k) = (3, -2)
- Radius: r = 4
- h: 3
- k: -2
Calculation Steps:
- Standard: (x - 3)² + (y - (-2))² = 4²
- Simplify: (x - 3)² + (y + 2)² = 16
- Expand for general form
- Result: x² + y² - 6x + 4y - 3 = 0
Standard Form: (x - 3)² + (y + 2)² = 16
General Form: x² + y² - 6x + 4y - 3 = 0
Circle at Origin
Center (0, 0), radius 7
x² + y² = 49
Unit Circle
Center (0, 0), radius 1
x² + y² = 1
Frequently Asked Questions
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