Equation of Circle

The equation of a circle with its center at (h, k) and a radius r is:

(x - h)² + (y - k)² = r²

Every point (x, y) on the circle satisfies it. This calculator takes the center, (X1, Y1), and any point on the circle, (X2, Y2), and finds the radius, which completes the equation, along with the diameter, area and circumference.

How it works

The radius is the distance from the center to the point on the circle:

r = √((X2 - X1)² + (Y2 - Y1)²)

From the radius:

• Diameter: d = 2 * r
• Area: A = π * r²
• Circumference: c = 2 * π * r

Example

A circle with its center at (2, -3) passing through (5, 1):

1. Radius:

   r = √((5 - 2)² + (1 + 3)²) = √25 = 5

2. The equation: with h = 2, k = -3 and r² = 25:

   (x - 2)² + (y + 3)² = 25

3. Diameter, area and circumference:

   d = 10, A = 78.54 and c = 31.42

The general form

Multiplying the brackets out gives the general form of the equation:

x² + y² + D * x + E * y + F = 0

For the example above, x² + y² - 4x + 6y - 12 = 0. To go back, the center is (-D / 2, -E / 2) and the radius r = √(D² / 4 + E² / 4 - F).

Frequently asked questions

How do you find the equation from the two ends of a diameter?

The center is the midpoint of the two ends, and the radius half the distance between them. For the ends (5, 8) and (-7, 4), the center is (-1, 6), and the radius √(36 + 4) = √40, so the equation is (x + 1)² + (y - 6)² = 40.

What is the equation of the unit circle?

x² + y² = 1: a circle with its center at the origin and a radius of 1.