Question

∆LMN is an equilateral triangle. LM = 14 cm. As shown in the figure, three sectors are drawn with vertices as centers 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

Given, 
ΔLMN is an equilateral triangle.
LM = 14 cm
Radius of the sectors (r) = 7 cm

(1) ΔLMN is an equilateral triangle. LM = 14 cm    ...(Given)       

Area of an equilateral triangle is ${\displaystyle \frac{\sqrt{3}}{4}{\left(\text{side}\right)}^2}$.

A(ΔLMN) = ${\displaystyle \frac{\sqrt{3}}{4}{\left(14\right)}^2}$

A(ΔLMN) = ${\displaystyle \frac{\sqrt{3}}{4}\times 196}$

A(ΔLMN) = ${\displaystyle \sqrt{3}\times 49}$

A(ΔLMN) = 1.732 × 49

A(ΔLMN) = 84.868 ≈ 84.87 cm²

∴ The area of ΔLMN is 84.87 cm².

(2) We know that, all the angles of the equilateral triangle are equal. Thus, ∠L = ∠M = ∠N = 60°.

Central angle (θ) = 60° 

Area of sector = ${\displaystyle \frac{\theta }{360}\times \pi r^2}$

= ${\displaystyle \frac{60}{360}\times \frac{22}{7}\times 7^2}$

= ${\displaystyle \frac{1}{6}\times 22\times 7}$

= ${\displaystyle \frac{11\times 7}{3}}$

= ${\displaystyle \frac{77}{3}}$

= 25.67 cm² 

∴ Area of one sector = 25.67 cm² 

(3) Total area of all the three sectors = 3 × Area of one sector

= 3 × 25.67
= 77.01 cm²

(4) Area of shaded region = A(∆LMN) – total area of all three sectors 

= 84.87 − 77.01

= 7.86 cm²

∴ Area of shaded region = 7.86 cm².

(1) A (∆LMN) = 84.87 cm²
(2) Area of any one of the sectors = 25.67 cm² 

(3) Total area of all the three sectors = 77.01 cm²

(4) Area of the shaded region = 7.86 cm²