Question

Semicircles are drawn outside by taking every side of regular hexagon as a diameter. The perimeter of hexagon is 60 cm. Find the area of complete figure formed as such. ($\mathrm{\pi }=3.14$) ($\sqrt{3}=1.73$)

Answer 1

Let the side of the regular hexagon be a cm.
Perimeter of the hexagon = 6 × a
$\Rightarrow$ 60 cm = 6 × a
$\Rightarrow$ a = 10 cm
∴ Side of the regular hexagon = 10 cm

Diameter of the semicircle = side of the regular hexagon = 10 cm
Radius of the semicircle = $\frac{10}{2}$ cm = 5 cm

Now, area of the semicircle = $\frac{\pi r^2}{2}=\frac{3.14\times 5^2}{2}=39.25{\mathrm{cm}}^2$

Now, area of six semicircles = 39.25 × 6 cm² = 235.5 cm²

Also, area of the hexagon = $\frac{3\sqrt{3}}{2}{a}^2=\frac{3\times 1.73}{2}\times {\left(10\right)}^2{\mathrm{cm}}^2=259.5{\mathrm{cm}}^2$

∴ Area of the figure = area of six semicircles + area of the regular hexagon
= (235.5 + 259.5) cm²
= 495 cm²