Equation Circle Diameter Endpoints Calculator

Background.

Everything you need to pin down a circle is hidden in the two ends of one of its diameters. The centre sits exactly halfway between them, and the radius is half the distance from one to the other. Once you have those two facts, the circle's equation in standard form, (x − h)² + (y − k)² = r², follows immediately.

This is one of the standard exercises in coordinate geometry because it ties together two tools taught just before it: the midpoint formula and the distance formula. Given endpoints (1, 3) and (7, −5), the midpoint is (4, −1), the endpoints are 10 units apart, and the circle is (x − 4)² + (y + 1)² = 25. The same reasoning appears outside the classroom whenever someone measures two opposite points on a round feature, such as the rim of a pipe or a bolt circle on a drawing, and needs its centre.

The condition that matters is that the two points really are opposite ends of a diameter. Any two points on a circle can be joined by a chord, and infinitely many circles of different sizes pass through the same pair of points. Only when the chord goes through the centre do the two points determine the circle on their own. If all you have is three points somewhere on the circumference, that is a different problem, solved by fitting a circle through all three.

The page returns the centre's coordinates and the radius; squaring the radius gives the right-hand side of the equation.

What is equation circle diameter endpoints calculator?

A diameter is a chord through the centre of a circle, so its endpoints lie on the circle directly opposite one another. On a coordinate plane, the centre is the midpoint of those endpoints, ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2), and the radius is half the straight-line distance between them. With centre (h, k) and radius r, every point (x, y) on the circle satisfies (x − h)² + (y − k)² = r².

How to use this calculator.

1. 01Enter the first endpoint's coordinates as x₁ and y₁ and the second's as x₂ and y₂. The order of the two points does not change the answer.
2. 02Check that the points are the ends of a diameter, not just any two points on the circle; the page assumes the segment between them passes through the centre.
3. 03Read the centre (h, k) and the radius r from the results.
4. 04Write the equation as (x − h)² + (y − k)² = r². A negative coordinate turns a minus into a plus, so a centre at (4, −1) gives (y + 1)². For an exact r², use ((x₂ − x₁)² + (y₂ − y₁)²) ÷ 4 rather than squaring a rounded radius.

The formula.

The centre is the average of the endpoints: h = (x₁ + x₂) ÷ 2 and k = (y₁ + y₂) ÷ 2. The radius is half the diameter's length, found with Pythagoras' theorem on the horizontal and vertical gaps: r = √((x₂ − x₁)² + (y₂ − y₁)²) ÷ 2. Because the gaps are squared, it makes no difference which point you call first or whether a gap is negative. If the two points coincide, the diameter has no length and no circle exists, so the page refuses identical endpoints. The radius is shown as a decimal; when the squared distance is not a perfect square, as with endpoints (0, 0) and (5, 3), r is irrational even though r² is an exact 8.5.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.