Circle Center Calculator

Background.

Finding the middle of something round is an old workshop problem. A woodturner needs the centre of a blank before mounting it on the lathe, a machinist needs the axis of a drilled bore, and a draughtsman may need the centre of an arc taken from a drawing. If you can identify two points that sit exactly opposite each other, the ends of a diameter, the rest is arithmetic: the centre lies halfway between them and the radius is half the distance across.

The hard part is being sure the two points really are opposite. Any two points on a circle are joined by a chord, and the centre lies somewhere on the perpendicular bisector of that chord, not necessarily at its midpoint. The points (0, 0) and (6, 8) sit on a circle centred at (3, 4) with radius 5, but they also sit on one centred at (−1, 7) with radius about 7.071. Only in the first case are they a diameter, and two points alone cannot tell the circles apart.

There are dependable ways to find a true diameter. Place the corner of a carpenter's square on the rim of a disc, and its two arms cross the rim again at the ends of a diameter, because any angle inscribed in a semicircle is a right angle. A machinist centring on a bore touches opposite walls along one axis, moves to the midpoint, then touches along the other axis; that second pair of touches lies on a diameter.

Enter both endpoints in the same unit and from the same origin, and you get the centre as an x and y pair with the radius in that unit. With three points on the rim and no known diameter, you need the circle through three points instead.

What is circle center calculator?

The centre of a circle is the one point at the same distance from every point on the circle. Every diameter passes through it and is cut exactly in half there. So when you know the two endpoints of a diameter, (x₁, y₁) and (x₂, y₂), the centre is their midpoint, whose coordinates are simply the averages of the two x values and the two y values. The radius is the distance from the centre to the rim, half the length of the diameter.

How to use this calculator.

1. 01Start from two points you know are diametrically opposite: marked on the object with a carpenter's square, read off a drawing, or given in a problem as the ends of a diameter.
2. 02Enter the first endpoint's x and y coordinates, then the second's. Negative coordinates are fine, and it makes no difference which point you call the first.
3. 03Measure both points from the same origin in the same unit, such as millimetres from one corner of a drawing or metres on a site grid; the radius is returned in that unit.
4. 04Read the centre's x and y and the radius. To confirm the points really were a diameter, take a third point on the rim and check that its distance from the centre equals the radius.

The formula.

The centre is h = (x₁ + x₂) ÷ 2 and k = (y₁ + y₂) ÷ 2, the midpoint formula, which averages each coordinate. The radius is r = √((x₂ − x₁)² + (y₂ − y₁)²) ÷ 2: the distance formula, which is Pythagoras' theorem applied to the horizontal and vertical gaps between the endpoints, halved because a diameter is twice the radius. No constants enter. Sliding both points by the same amount slides the centre by that amount and leaves the radius alone, since the radius depends only on the gaps between the points. If the two endpoints are the same point, the radius comes out as 0, a degenerate circle that signals an entry slip rather than a real result. With h, k and r in hand, the circle's equation is (x − h)² + (y − k)² = r².

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function 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.