Answer
3.828
Work Step by Step
Step 1. Draw a diagram as shown in the figure, the three sides of the triangle (connecting centers of circles) are 9,10,11.
Step 2. Use the Law of Cosines to calculate angle C in radians, $11^2=10^2+9^2-2\times9\times10cosC$ which gives $cosC=\frac{1}{3}$ and $\angle C\approx1.231$
Step 3. Repeat Step 2 for other angles to get $\angle B\approx1.030$ and $\angle A\approx0.881$
Step 4. Use Heron's Formula to calculate the are of the triangle: $s=\frac{11+10+9}{2}=15$, $A=\sqrt {15(15-11)(15-10)(15-9)}\approx42.426$
Step 5. Use the formula $A_f=\frac{r^2\theta}{2}$ to calculate the three fan areas inside each circle and the triangle:
$A1=\frac{6^2\times0.881}{2}=15.858$, $A2=\frac{5^2\times1.030}{2}=12.875$, $A3=\frac{4^2\times1.231}{2}=9.848$
Step 6. The area between the circles is given by $A-A1-A2-A3=42.426-15.858-12.875-9.848=3.828$
Note: depends on the accuracy during the calculation, there could be a small difference in the final result.
