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Inner area



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The three radii added pairwise are the sides of a triangle defined by the centers of the circles.

The angles are given by law of cosines.

The chords are given by the radii and angles, as are the segment areas.

The area of the triangle formed by the three points of tangency is given by the chord lengths and Heron's formula.

The area bounded by the circles is that area minus the three segment areas.


I let each radius take the values 1,2,3,4,5 to obtain 125 bounded areas.

These ranged froim 0.161254 for (1,1,1) to 4.03162 for (5,5,5).

The median value (number 63 in the ordered list) was found to be 0.95248.


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Assuming yes to both questions:


Assign a range of values to the radii.

Let each have discrete values equally spaced over the interval [1, 5].

For example 1.5, 2.5, 3.5, 4.5, which would give 64 cases.


Compute the areas and find the median value.


The circles' centers define a triangle whose sides are pairwise sums of the radii.

From the sides calculate the triangle's area.

Get the three angles from the law of cosines.

Each circle has a sector of area  A r2/2 where A is the angle for that circle.

Subtract the area of the three sectors from that of the triangle.



It's NOT the area for three circles having median radius of 3.
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