How to use the circle from three points calculator
- Enter the x and y coordinates of the first point.
- Enter the coordinates of the second point and the third point. They can be in any order and any unit.
- Read the center and radius of the circle through all three points, along with the diameter, circumference, area and equation.
- Check the table at the bottom: the distance from the center to each point should be the same.
Formula
D = 2 x (x1 (y2 - y3) + x2 (y3 - y1) + x3 (y1 - y2))
center x = ((x1^2 + y1^2)(y2 - y3) + (x2^2 + y2^2)(y3 - y1) + (x3^2 + y3^2)(y1 - y2)) / D
center y = ((x1^2 + y1^2)(x3 - x2) + (x2^2 + y2^2)(x1 - x3) + (x3^2 + y3^2)(x2 - x1)) / D
radius = distance from the center to any of the points
The center is where the perpendicular bisectors of the segments between the points meet. If D is zero, the points lie on one line and no circle exists.
Three points on a circle
For the points (1, 7), (6, 6) and (7, 1), D = 2 x (1 x 5 + 6 x (-6) + 7 x 1) = -48 and the center works out to (2.75, 2.75). Each point is 4.5962 away from it, so the radius is 4.5962, the circumference is 28.88 and the area is 66.37. The equation is (x - 2.75)^2 + (y - 2.75)^2 = 21.125.
For (0, 0), (4, 0) and (0, 3), the center is (2, 1.5) and the radius is 2.5. That is the 3, 4, 5 right triangle, whose hypotenuse of 5 is a diameter, as the geometry of a right triangle inscribed in a circle says it must be.
Uses
Finding the circle through three points helps when you need the radius of a curved wall, a wheel, a table edge or a part from three measured points, or to complete an arc from a few marks. If you know the width of an arch and its height instead, the Arch Template Calculator gives the radius and the heights along the span.
For the circumference, area and arc of a circle you already know the radius of, use the Circle Calculator.
Points that do not work
If the three points are on one straight line, or two of them are the same point, there is no circle through them, and the tool says so instead of returning a huge radius. Points that are nearly in a line give a very large radius, so measure them carefully.
More shape tools
For other shape and measurement questions, try the Polygon Area Calculator, Longest Length Around a Corner Calculator and Bike Spoke Length Calculator.
Assumptions and limits
- The points are on a flat plane and given as x and y coordinates.
- All three points are different and do not lie on one line.
- Lengths come out in the same unit as the coordinates.
- Results are shown to four decimal places.
Frequently asked questions
How do you find a circle from three points?
Find where the perpendicular bisectors of two of the segments between the points cross. That is the center, and its distance to any point is the radius.
Can any three points make a circle?
Any three points that are not in a straight line lie on exactly one circle. Three points on a line have no circle through them.
What is the equation of the circle?
It is (x - cx)^2 + (y - cy)^2 = r^2, with the center and radius the tool finds. The page prints it with your numbers.
What unit is the radius in?
The same unit as your coordinates. If the coordinates are inches, the radius is in inches and the area is in square inches.