📐 Equation of a Circle Calculator
Find the equation of a circle
How to Use This Calculator
Choose Input Method
Select whether you know the center and radius, or center and a point on the circle.
Enter Center Coordinates
Input the x and y coordinates of the circle's center (h, k).
Enter Radius or Point
If using center-radius, enter the radius. If using center-point, enter coordinates of a point on the circle.
Calculate
Click "Calculate Equation" to get both standard and general form equations.
Formula
(x - h)² + (y - k)² = r²
Standard form of a circle equation
Where:
- (h, k) = center of the circle
- r = radius of the circle
- (x, y) = any point on the circle
General Form:
x² + y² + Dx + Ey + F = 0
Where D = -2h, E = -2k, F = h² + k² - r²
Example 1: Center at (0, 0), radius = 5
Standard form: (x - 0)² + (y - 0)² = 5²
Simplified: x² + y² = 25
Example 2: Center at (2, -3), radius = 4
Standard form: (x - 2)² + (y + 3)² = 4²
Expanded: (x - 2)² + (y + 3)² = 16
Finding Radius from Center and Point:
r = √[(x - h)² + (y - k)²]
Distance formula from center to point
About Equation of a Circle Calculator
The Equation of a Circle Calculator finds the mathematical equation that describes a circle. Circles can be represented in standard form (x-h)² + (y-k)² = r² or general form x² + y² + Dx + Ey + F = 0.
When to Use This Calculator
- Geometry: Solve circle problems in coordinate geometry
- Algebra: Find circle equations for algebraic problems
- Education: Learn and practice circle equation concepts
- Engineering: Calculate circular paths and trajectories
- Computer Graphics: Define circles for rendering and design
- Verification: Check manual calculations of circle equations
Why Use Our Calculator?
- ✅ Multiple Methods: Works with center-radius or center-point
- ✅ Both Forms: Shows standard and general form equations
- ✅ Instant Results: Calculate equations immediately
- ✅ Educational: Displays formulas and calculation steps
- ✅ Accurate: Precise mathematical calculations
- ✅ 100% Free: No registration required
Understanding Circle Equations
The standard form (x-h)² + (y-k)² = r² directly shows the center (h,k) and radius r. The general form is useful for solving systems of equations and finding intersections.
- Standard form is easier to read: you can immediately see center and radius
- General form is useful when solving multiple equations simultaneously
- You can convert between forms using algebraic manipulation
- The equation represents all points equidistant from the center
Real-World Applications
Navigation: Calculate circular paths for GPS routing and waypoint planning.
Physics: Model circular motion, orbits, and rotational systems.
Computer Graphics: Render circles, arcs, and curved shapes using circle equations.
Architecture: Design circular structures and verify circular dimensions.
Frequently Asked Questions
What is the standard form of a circle equation?
Standard form is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form makes it easy to identify the center and radius directly.
How do I convert from general form to standard form?
Complete the square for both x and y terms. For x² + Dx, add (D/2)². For y² + Ey, add (E/2)². Then rearrange to get (x-h)² + (y-k)² = r² form.
Can the center be at the origin?
Yes! If center is at (0, 0), the equation simplifies to x² + y² = r². This is a circle centered at the origin with radius r.
What if I have three points on the circle?
You can find the circle by solving a system of equations. Substitute each point into (x-h)² + (y-k)² = r² and solve for h, k, and r. This calculator supports this method.
Can the radius be negative?
No, radius is always positive. It represents a distance, which cannot be negative. If you get a negative value in calculations, take the absolute value.
How do I check if a point is on the circle?
Substitute the point's coordinates into the circle equation. If (x-h)² + (y-k)² = r², the point is on the circle. If less than r², it's inside; if greater, it's outside.