Equation of a Circle from Diameter Endpoints
Enter the coordinates of the two endpoints of a circle's diameter and instantly get the standard-form equation, center, radius, area and circumference, with a coordinate plot.
- 1
Center h (midpoint x)
(1 + 5) ÷ 2 = 3 - 2
Center k (midpoint y)
(0 + 4) ÷ 2 = 2 - 3
Diameter
√((5 − 1)² + (4 − 0)²) = 5.6569Distance formula applied to the two diameter endpoints. - 4
Radius r
5.6569 ÷ 2 = 2.8284
How does this calculator work?
Given diameter endpoints (x₁,y₁) and (x₂,y₂): center h=(x₁+x₂)/2, k=(y₁+y₂)/2; radius r=√((x₂−x₁)²+(y₂−y₁)²)/2; equation (x−h)²+(y−k)²=r². Enter the two points to get the equation, center, radius, area and circumference.
Formula
How this is calculated
When the two endpoints of a circle's diameter are known, the center lies exactly at their midpoint by definition: h = (x₁+x₂)/2 and k = (y₁+y₂)/2. The radius equals half the length of the diameter, computed using the distance formula: r = √((x₂−x₁)²+(y₂−y₁)²) / 2. These two quantities fully determine the circle.
Substituting the center and radius into the standard form (x−h)²+(y−k)²=r² gives the equation of the circle. The standard form is convenient because h, k and r can be read off directly — any point (x,y) satisfying the equation lies exactly r units from the center, which is the geometric definition of a circle.
The calculator also reports the diameter (2r), circumference (2πr) and area (πr²), and plots the circle on a coordinate grid with the two diameter endpoints (P₁, P₂) and the center (C) marked. Inputs may be any real numbers — negative or fractional coordinates are fine.
Frequently asked questions
The center of a circle is equidistant from every point on the circumference. The two endpoints of a diameter are both on the circumference and lie on exactly opposite sides of the center, so by symmetry the center is halfway between them — i.e., the midpoint.
Expand (x−h)²+(y−k)²=r² to x²−2hx+h²+y²−2ky+k²=r², then collect constant terms: x²+y²+(−2h)x+(−2k)y+(h²+k²−r²)=0. This is the general form x²+y²+Dx+Ey+F=0 where D=−2h, E=−2k, F=h²+k²−r².
If x₁=x₂ and y₁=y₂ the diameter length is zero, giving r=0, which is degenerate — there is no valid circle. The calculator requires r>0 and shows a warning otherwise.
Also known as
TG we-Calculate Editorial Team. (2026). Equation of a Circle from Diameter Endpoints [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/equation-circle-diameter-endpoints-calculator
TG we-Calculate Editorial Team. "Equation of a Circle from Diameter Endpoints." TG we-Calculate. 2026. https://we-calculate.com/calculator/equation-circle-diameter-endpoints-calculator.
TG we-Calculate Editorial Team, "Equation of a Circle from Diameter Endpoints," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/equation-circle-diameter-endpoints-calculator
@misc{wecalculate_equation_circle_diameter_endpoints_calculator, title = {Equation of a Circle from Diameter Endpoints}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/equation-circle-diameter-endpoints-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?