Circle Equation from Diameter Endpoints

Find the standard form equation of a circle given two endpoints of a diameter, with step-by-step solutions.

Enter Diameter Endpoints

Endpoint A (x1, y1)
Endpoint B (x2, y2)

Result

Circle Equation (Standard Form)
(x-4)^2 + (y-5)^2 = 18
standard form
Center (h, k) (4, 5)
Radius (r) 4.242641
Radius Squared (r2) 18
Diameter (d) 8.485281
Circumference 26.656461
Area 56.548668
General Form x^2 + y^2 - 8x - 10y + 23 = 0

Step-by-Step Solution

(x-h)^2 + (y-k)^2 = r^2

Finding a Circle Equation from Diameter Endpoints

When you are given two points that are endpoints of a diameter of a circle, you can determine the complete equation of the circle. The process involves two main steps: finding the center (which is the midpoint of the diameter) and finding the radius (which is half the length of the diameter).

The standard form of a circle equation is (x - h)2 + (y - k)2 = r2, where (h, k) is the center and r is the radius.

Related Formulas

Standard Form

The most common way to write a circle equation, showing center and radius directly.

(x-h)^2 + (y-k)^2 = r^2

General Form

Expanded form with coefficients. Useful for algebraic manipulation.

x^2 + y^2 + Dx + Ey + F = 0

Center (Midpoint)

The center is the midpoint of the diameter endpoints.

C = ((x1+x2)/2, (y1+y2)/2)

Radius

The radius is half the distance between the diameter endpoints.

r = d/2 = sqrt((x2-x1)^2+(y2-y1)^2)/2

Circumference

The perimeter or total arc length of the circle.

C = 2*pi*r = pi*d

Area

The total region enclosed by the circle.

A = pi*r^2

Step-by-Step Method

  1. Find the center: Calculate the midpoint of the two diameter endpoints using h = (x1 + x2)/2 and k = (y1 + y2)/2.
  2. Find the diameter length: Use the distance formula d = sqrt((x2 - x1)2 + (y2 - y1)2).
  3. Find the radius: Divide the diameter by 2: r = d/2.
  4. Write the equation: Substitute h, k, and r into (x - h)2 + (y - k)2 = r2.

Converting Between Standard and General Form

To convert from standard form to general form, expand the squares: (x - h)2 + (y - k)2 = r2 becomes x2 - 2hx + h2 + y2 - 2ky + k2 = r2. Rearranging: x2 + y2 - 2hx - 2ky + (h2 + k2 - r2) = 0. Here D = -2h, E = -2k, and F = h2 + k2 - r2.

Practical Applications

  • Architecture: Designing circular arches, domes, and windows requires knowing the center and radius from given constraints.
  • Astronomy: Modeling circular orbits from observed positions of celestial bodies.
  • CAD/Engineering: Defining circular features in technical drawings from reference points.
  • Navigation: Determining circular search areas centered between two known positions.
  • Computer Graphics: Rendering circles and arcs from control points in graphic design software.

Worked Example

Given diameter endpoints A(2, 3) and B(8, 7), find the circle equation.

  1. Center: h = (2+8)/2 = 5, k = (3+7)/2 = 5. So center = (5, 5).
  2. Diameter: d = sqrt((8-2)2 + (7-3)2) = sqrt(36 + 16) = sqrt(52).
  3. Radius: r = sqrt(52)/2, so r2 = 52/4 = 13.
  4. Equation: (x - 5)2 + (y - 5)2 = 13.