3. Three equal circles of radius $r$ each pass through the centres of the other two. What is the area of intersection that is common to all the three circles? Explain your reasoning.
Soln.
Let the centres of the circles be $\mathrm{O} 1, \mathrm{O} 2, \mathrm{O} 3$.
Let A be the area of the triangle $\Delta \mathrm{O} 1 \mathrm{O} 2 \mathrm{O} 3$ and let $\mathrm{S}$ be the area of the sector of the circle with centre $\mathrm{O} 1$ and bound by the points $\mathrm{O} 2$ and $\mathrm{O} 3$.
[1 mk]

The desired area is $\mathrm{A}+3(\mathrm{~S}-\mathrm{A}) \quad[1 \mathrm{mk}]$
$$
\begin{array}{l}
=3 \mathrm{~S}-2 \mathrm{~A} \\
=3 \times \frac{1}{6} \pi r^{2}-2 \times \frac{1}{4} \sqrt{3} r^{2} \\
=\frac{1}{2} r^{2}(\pi-\sqrt{3}) \quad[1 \mathrm{mk}]
\end{array}
$$