Question

There are two concentric hexagons. Each of the side of both the hexagons are parallel. Each side of internal regular hexagon is 8 cm. What is the area of the shaded region, if the distance between the corresponding parallel sides is 2√3 cm?

Answer 1

In the second figure, ΔOPQ is an equilateral triangle as a regular hexagon is divided into six equilateral triangles. ∴ ∠OPC = 60º

⇒ \(\frac{OC}{OP}\) = sin 60° ⇒ OC = OP sin 60° ⇒ OC = 8 x \(\frac{\sqrt3}{2}\) cm = 4√3 cm.

Now in similar Δs OPC and OAD, \(\frac{OC}{OD}\) = \(\frac{OP}{OA}\) ⇒ \(\frac{4\sqrt3}{6\sqrt3}\) = \(\frac{8}{OA}\)

⇒ OA = 12 cm ⇒ AB = OA = 12 cm

(∵ OD = OC + 2√3 = 4√3  + 2√3 = 6√3 cm)

∴ Required area = Area of outer hexagon – Area of inner hexagon

= \(\frac{3\sqrt3}{2}\)(12² - 8²) cm² = 120√3 cm².