Two Circles Calculator

Name: Two Circles - Geometry Calculator
Author: Jürgen Kummer

Calculations at two intersecting circles with the same size. The intersection of both circles is a pointed oval. The lateral overhangs, that is, what belongs to one circle but not to both, are crescents. For circles with different sizes, see lune. When more circles intersect, further shapes are created, for this, see three circles. The three-dimensional equivalent of two intersecting circles of equal size is the double sphere.
Enter radius and distance and choose the number of decimal places. Then click Calculate. The distance of the circle centers must be smaller than their diameter (twice the radius).

Formulas:

b = a + 2 r

h = 2 r - a

l = 2 \cdot \sqrt{r \cdot h - \frac{h^{2}}{4}}

p = 4 π r - 4 r \cdot \arccos (1 - \frac{h}{2 r})

A = 2 π r^{2} - \frac{4 r^{2} \cdot \arccos (1 - \frac{h}{2 r})}{2} + l \cdot (r - \frac{h}{2})

pi:

π = 3.141592653589793 ...

Radius, distance, width, height, length and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The calculation for this shape is done using the formulas for a circle and those for a circular segment. The perimeter is that of two circles minus the lengths of two segment arcs, as these lie within the shape and therefore do not count to the perimeter. The area is that of two circles minus that of two circular segments. These two circular segments, together the pointed oval, are the area where the two circles intersect. They overlap and are therefore subtracted once to avoid being counted twice.

Two intersecting circles are often used as patterns or in design. Their use in set theory is well-known, showing that objects can belong to one or both sets. Here, the circles represent set A and set B. If both circles are filled, this represents the union of the sets. In this case, the elements are in either A or B. If only the intersection is filled, then the elements are in both A and B. If only one circle is filled, this represents set A excluding B, or B excluding A. If both circles are filled, but not the intersection, this represents A xor B, where xor is the exclusive or.

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