Alright, let’s see:
Get the area of the triangle that forms between their center points. That part should be easy enough, yes? Using your example of the 2, 3, and 4 radii, we can get a triangle with leg lengths of 5, 6, and 7.
We use Heron’s formula (i admit I had to look up the name, but I did know it existed!) which says: Area = sqrt(s(s-a)(s-b)(s-c))
In this case, a, b, and c are the leg lengths, and s is 1/2(a+b+c). So, doing all that math, the area of the triangle is 14.69 (heh…69)
Next we need to find the area of the arc segments of each circle contained within the triangle. For this, we need each angle in the triangle. I’m sure there’s a simpler way to do this, but the best way I know is the law of sines:
a/sin A = b/sin B = c/sin C = 2R.
R is the radius of the circumcircle, so 2R is the diameter. As you might guess, the circumcircle is the circle formed around the triangle with the three points on the circle. The diameter of the circumcircle is:
abc/2*Area. We know all of these, so the diameter is 7.15.
Finally, we can get the angles. The angle named for each leg is the one opposite that leg, so first up:
a/sin A = 7.15 = 5/sin A.
Arcsin A = 5/7.15 = 44.37 degrees. Since length a was the smallest length, it was opposite the angle formed in the “4” triangle.
b/sin B = 7.15 = 6/Sin B
Arcsin B = 6/7.15 = 57.05 degrees, which is the angle in the “3” triangle.
c/sin C = 7.15 = 7/sin C
Arcsin C = 7/7.15 = 78.24 degrees.
Let me check that those angles add up to 180…yup (well, close enough since I rounded.)
So, now we can get the area of the arc segments.
Circle “2”: Area = 4Pi, arc segment area = 4Pi(78.24/360) = 0.87Pi
Circle “3”: Area = 9Pi, arc segment area = 9Pi(57.05/360) = 1.43Pi
Circle “4”: Area = 16Pi, arc segment area = 16Pi(44.37/360) = 1.97Pi
So, the total area of all the arc segments is 4.27Pi = 13.41.
From before, the area of the triangle is 14.69.
14.69-13.41 = 1.28
1.28
I pray I did all my math right…