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A plane contains points A and B with AB = 1. Let AB =1. Let S be the union of all disk of radius 1 in the plane that covers $\overline{AB}$ . What is the area of S ?

I have the following options :

(a) $2\pi +\sqrt{3}$

(b) $\frac{8\pi}{3}$

(c) $3\pi -\frac{\sqrt3}{2}$

(d) $\frac{10\pi}{3}-\sqrt{3}$

Please suggest how to proceed.

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2 Answers

Hint: Draw the two equilateral triangles that have $AB$ as one of their sides. Call them $ABC$ and $ABC'$.

We work with the top one. But the same has to be done with the bottom one.

Draw the circular arc with centre $A$ passing through $B$ and $C$. Do the same, centre $B$, passing through $A$ and $C$.

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Hint: Here is a diagram. Each dotted angle is $\frac{2\pi}{3}$ or $\frac13$ of a circle. The circles have radius $1$ and the arcs have radius $2$. The diamond shape in the middle has height $1$ and width $\sqrt3$.

$\hspace{4.5cm}$enter image description here

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