4. In Figure 1, the three shaded areas are formed by two circles with a radius of 1 and their external common tangents. If the area of the middle shaded part is equal to the sum of the areas of the upper and lower shaded parts, then the length of the common chord of the two circles is ( ).
(A) $\frac{\sqrt{5}}{2}$
(B) $\frac{\sqrt{6}}{2}$
(C) $\frac{1}{2} \sqrt{25-\pi^{2}}$
(D) $\frac{1}{2} \sqrt{16-\pi^{2}}$
4.D.

Let the distance between the centers be $d$. From the condition, it is easy to know that the area of rectangle $A B C D$ equals the area of one circle, i.e., $2 d=\pi$, so,
$$
\begin{array}{l}
O_{1} P=\frac{d}{2}=\frac{\pi}{4}, \\
M N=2 M P=2 \sqrt{1-\left(\frac{\pi}{4}\right)^{2}}=\frac{1}{2} \sqrt{16-\pi^{2}} .
\end{array}
$$