Area Between Three Circles of Differing Radii From the link in wikipedia 

http://web.gnowledge.org/wiki/index.php/Area_Between_Three_Circles_of_Differing_Radii

OPEN QUESTION:

What is the equation, in three variables, relating the radii of three circles to the area between them, when each is tangent to the other two?



But wiki says that it is an open question  and I am interested what is main reason of it? or is the author simply trying to understand if anybody can solve it (while it's answer is known?)
As a follow-up to Gerry's answer and Day Late Don's comment there, let's consider the triangle formed by the centers of the circles and subtract the sectors.  Call the circles 1, 2, and 3, with radii $r_1$, $r_2$, and $r_3$, respectively, and let $\theta_1$, $\theta_2$, and $\theta_3$, be the interior angles of the triangle formed by the centers of the circles, at the centers of circles 1, 2, and 3, respectively.

Applying the Law of Cosines (and simplifying some) to find $\theta_1$, $\theta_2$, and $\theta_3$ gives:
$$\theta_1=\arccos\left(1-\frac{2r_2r_3}{(r_1+r_2)(r_1+r_3)}\right)$$
$$\theta_2=\arccos\left(1-\frac{2r_1r_3}{(r_2+r_1)(r_2+r_3)}\right)$$
$$\theta_3=\arccos\left(1-\frac{2r_1r_2}{(r_3+r_1)(r_3+r_2)}\right)$$
so the sector areas are
$$\frac{1}{2}\theta_1r_1^2=\frac{1}{2}r_1^2\arccos\left(1-\frac{2r_2r_3}{(r_1+r_2)(r_1+r_3)}\right)$$
$$\frac{1}{2}\theta_2r_2^2=\frac{1}{2}r_2^2\arccos\left(1-\frac{2r_1r_3}{(r_2+r_1)(r_2+r_3)}\right)$$
$$\frac{1}{2}\theta_3r_3^2=\frac{1}{2}r_3^2\arccos\left(1-\frac{2r_1r_2}{(r_3+r_1)(r_3+r_2)}\right)$$

Applying Hero's formula to find the area of the triangle (and simplifying) gives $$K_\triangle=\sqrt{r_1r_2r_3(r_1+r_2+r_3)}.$$

So the area of the target region is
$$\sqrt{r_1r_2r_3(r_1+r_2+r_3)}-\frac{1}{2}\left(\begin{align}
r_1^2&amp;\arccos\left(1-\frac{2r_2r_3}{(r_1+r_2)(r_1+r_3)}\right)
\\&amp;+r_2^2\arccos\left(1-\frac{2r_1r_3}{(r_2+r_1)(r_2+r_3)}\right)
\\&amp;+r_3^2\arccos\left(1-\frac{2r_1r_2}{(r_3+r_1)(r_3+r_2)}\right)
\end{align}\right).$$
In the special case of all 3 circles of the same radius $r$, the area is given by the equation $k r^2$  where $k=0.161254481$.