Question

$∆$ LMN is an equilateral triangle. LM = 14 cm. As shown in figure, three sectors are drawn with vertices as centres and radius 7 cm.

Find,

(1) A ($∆$LMN)

(2) Area of any one of the sectors .

(3) Total area of all the three sectors.

(4) Area of the shaded region.

Answer 1

∆LMN is an equilateral triangle.

∴ LM = MN = LN = 14 cm

∠L = ∠M = ∠N = 90º

(1)
Area of ∆LMN = $\frac{\sqrt{3}}{4}{\left(\mathrm{Side}\right)}^2=\frac{\sqrt{3}}{4}\times {\left(14\right)}^2=\frac{1.732}{4}\times 196$ = 84.87 cm²

(2)
Radius of the each sector, r = 7 cm

Area of any one of the sectors = $\frac{\theta }{360°}\times \mathrm{\pi }{r}^2=\frac{60°}{360°}\times \frac{22}{7}\times {\left(7\right)}^2$ = 25.67 cm²

(3)
Total area of all the three sectors = 3 × Area of any one of the sectors = 3 × 25.67 = 77.01 cm²

(4)
Area of the shaded region = Area of ∆LMN − Total area of all the three sectors = 84.87 − 77.01 = 7.86 cm²