Find the intersection of two circles

To use the calculator, enter the x and y coordinates of a center and radius of each circle.

A bit of theory can be found below the calculator.

Intersection of two circles

First Circle

x

y

radius

Second circle

x

y

radius

Calculation precision

Digits after the decimal point: 2

Calculate

Distance analysis

 

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Circles intersection

The task is relatively easy, but we should take into account the edge cases – therefore we should start by calculating the cartesian distance d between two center points, and checking for edge cases by comparing d with radiuses r1 and r2.

Here are the possible cases (distance between centers is shown in red):

Case	Description	Rule
	Trivial case: the circles are coincident (or it is the same circle)
	The circles are separate
	One circle is contained within the other
	Two intersection points	You have one or two intersection points if all rules for the edge cases above are not applied
	One intersection point	Trivial case of two intersection points

So, if it is not an edge case, to find the two intersection points, the calculator uses the following formulas (mostly deduced with Pythagorean theorem), illustrated with the graph below:

The first calculator finds the segment a

and then the segment h

To find point P3, the calculator uses the following formula (in vector form):

And finally, to get a pair of points in case of two points intersecting, the calculator uses these equations:
First point:

Second point:

Note the opposite signs before the second addend

For more information, you can refer to Circle-Circle Intersection and Circles and spheres