Circles $A,B$ each have radius $1$ and share a point of tangency. Circle $C$ has radius $\sqrt3$ and is tangent to line $AB$ at the midpoint of $\overline{AB}$. What is the area inside circle $C$ but outside circles $A$ and $B$?

1. Place $A=(-1,0)$, $B=(1,0)$, $M=(0,0)$; then $C=(0,\sqrt3)$. Let $P$ be an intersection point of circles $A$ and $C$. Then $AC=\sqrt{1^2+(\sqrt3)^2}=2$, and $AP^2+CP^2=1+3=AC^2$, so $\angle APC=90^\circ$.
2. Let $\alpha=\angle PAC$ and $\beta=\angle PCA$. Then $\tan\alpha=\frac{CP}{AP}=\sqrt3$, so $\alpha=60^\circ=\frac\pi3$ and $\beta=30^\circ=\frac\pi6$.
3. The overlap of circles $A$ and $C$ is two circular sectors minus the kite $APCP'$: $$1^2\cdot\frac\pi3+(\sqrt3)^2\cdot\frac\pi6-2\cdot\tfrac12\cdot1\cdot\sqrt3=\frac{5\pi}6-\sqrt3.$$
4. Circles $A$ and $B$ meet only at $M$, so the two overlaps with $C$ do not overlap each other. The desired area is $$3\pi-2\left(\frac{5\pi}6-\sqrt3\right)=\boxed{\frac{4\pi}3+2\sqrt3}\approx7.65.$$