About this calculator
How to quickly find the standard equation or general equation of a circle? The circle is one of the most basic figures in plane geometry. The equation of a circle has two commonly used forms: the standard equation (x-a)²+(y-b)²=r² and the general equation x²+y²+Dx+Ey+F=0. Among them (a, b) are the coordinates of the center of the circle, and r is the radius.
In practical problems, it is often necessary to convert between the two forms, or to find the equation of a circle based on known conditions. For example, if the center and radius of a circle are known, the standard equation can be written directly. Given three points, the equation of the circle can be found through a system of simultaneous equations.
Equations of circles are widely used in engineering design, computer graphics, physics and other fields. In mechanical design, the outline of a circular part is described by the equation of a circle. In computer graphics, drawing a circle requires the equation of the circle.
Our circle equation calculator can find the equation of a circle based on different known conditions and convert between standard equations and general equations. Supports multiple input methods and provides detailed calculation steps and geometric illustrations.
What it calculates
The circle equation calculator builds a circle equation from center and radius, or helps identify center and radius from a general equation.
Formula
A circle with center (h, k) and radius r has equation (x - h)^2 + (y - k)^2 = r^2.
Inputs
- Center coordinates h and k.
- Radius r.
- Or coefficients from a general circle equation.
Example
| Center | Radius | Equation |
|---|---|---|
| (0, 0) | 5 | x^2 + y^2 = 25 |
| (2, -3) | 4 | (x - 2)^2 + (y + 3)^2 = 16 |
| (-1, 1) | 2 | (x + 1)^2 + (y - 1)^2 = 4 |
How to interpret the result
A circle equation describes all points whose distance from the center equals the radius. The radius must be nonnegative; larger radius means a larger circle.
Common mistakes
- Signs of h and k are easy to reverse in standard form.
- The radius cannot be negative.
- Complete the square before reading center and radius from general form.
How to use
Using the Circle Equation Calculator is very easy. Just select the known conditions and enter the parameters.
**Method 1: Known center and radius** Input the center point (a, b) and radius r, and directly get the standard equation (x-a)²+(y-b)²=r².
**Example 1:** Circle center (2,3), radius 5. Equation: (x-2)²+(y-3)²=25.
**Method 2: Three known points** Enter the coordinates of three points and the calculator solves the equation of the circle.
**Example 2:** Circle passing through the points (0,0), (4,0), (0,3). Assume the equation x²+y²+Dx+Ey+F=0, substitute three points into the system of equations, and solve it to get D=-4, E=-3, F=0.
**Method 3: Convert standard equation to general equation** Expand (x-a)²+(y-b)²=r², we get x²+y²-2ax-2by+(a²+b²-r²)=0.
Main features
• Multiple inputs: circle center radius, three points, two points plus radius, etc. • Two-way conversion: standard equation ↔ general equation • Properties of circles: automatically calculate center, radius, area, and circumference • Positional relationship: Determine the positional relationship between a point and a circle, a straight line and a circle • Geometric Diagram: Draw the shape of a circle • Calculation steps: show detailed solution process • Equation verification: Verify whether the point is on the circle • Tangent equation: find the equation of the tangent line passing through a point on the circle • Batch calculation: supports calculation of multiple circles • Totally free: no registration required, use anytime
Use cases
• Analytical geometry learning: students learn the equation of a circle • Engineering Design: Design circular parts and trajectories • Computer graphics: drawing circles and arcs • Physics: Analyze circular motion • Architectural design: designing circular structures • GIS: processing circular areas • Exam Prep: Quickly Solve the Equation of a Circle • Teaching aid: teacher explains the equation of a circle • Mechanical design: Calculation of circular part parameters • Game development: Implementing circular collision detection