In the given figure, ∆𝑃𝑄𝑅 is an equilateral triangle of side 8 𝑐𝑚 and 𝐷, 𝐸, 𝐹 are centres of circular arcs, each radius 4 𝑐𝑚. Find the area of shaded region. (Use π = 3. 14 and  $\sqrt{3}$     = 1. 732)

 

We need to find the area of shaded region provided that ∆𝑃𝑄𝑅 is an equilateral triangle of side 8 𝑐𝑚 and 𝐷, 𝐸, 𝐹 are centres of circular arcs, each radius 4 𝑐𝑚.

It can be observed that the arcs form the sector in the triangle and hence, the area of shaded region is

𝐴𝑟𝑒𝑎 𝑜𝑓 𝑠ℎ𝑎𝑑𝑒𝑑 𝑟𝑒𝑔𝑖𝑜𝑛 = 𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝑃𝑄𝑅 − 3(𝐴𝑟𝑒𝑎 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟)

⇒ 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑠ℎ𝑎𝑑𝑒𝑑 𝑟𝑒𝑔𝑖𝑜𝑛 = $\frac{\sqrt{3}}{4}\times sid{e}^2 - 3 \times \left(\frac{\theta }{360°}\times {\mathrm{\pi r}}^2\right)$

⇒ 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑠ℎ𝑎𝑑𝑒𝑑 𝑟𝑒𝑔𝑖𝑜𝑛 = $$\begin{gathered} \frac{\sqrt{3}}{4}\times 8^2 - 3 \times \left(\frac{60°}{360°}\times \pi 4^2\right) \\ 16\sqrt{3} - 8\mathrm{\pi } \\ 16 \times 1.732 - 8 \mathrm{x} 3.14 \\ 27.712 - 25.12 \\ 2.592 {\mathrm{cm}}^2 \end{gathered}$$

Hence, the area of the shaded region is  2.592 cm².