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 `sqrt3/4 ("side")^2`.

A(ΔLMN) = `sqrt3/4 (14)^2`

A(ΔLMN) = `sqrt3/4 × 196`

A(ΔLMN) = `sqrt3 × 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 = `θ/360 × πr^2`

= `60/360 × 22/7 × 7^2`

= `1/6 × 22 × 7`

= `(11 × 7)/3`

= `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