Question Number 179811 by Acem last updated on 02/Nov/22

Commented byAcem last updated on 02/Nov/22

∗ ℓ is an arc ∗ we can get 2 equilateral triangles formed by it vertices

Commented byAcem last updated on 02/Nov/22

Find the area of the equilateral triangles formed by the vertices of this hexagon

Commented bymr W last updated on 02/Nov/22

Commented bymr W last updated on 02/Nov/22

area of equilateral (red) is half of the area of the hexagon: (6/2)×(((√3)l^2 )/4)=((3(√3))/4)((π/( (√3))))^2 =(((√3)π^2 )/4)

Answered by Acem last updated on 05/Nov/22

We will solve it in two cases: A) 2 seperated right triangles B) Area of union of 2 triangles ♮hexagonal starε L_H = r = (ℓ/θ) ; θ= (π/3) ⇒ L_H = (√3) A• Area_(2 triang.) = Area_(Hexag.) = ((3(√3))/2) L_H ^2 = ((9(√3))/2) un^2 B• Area_(hex. star) = Area_(2 triang.) − Area_(Small hexagon) ...(1) ; Small hexagon = common area of 2 right trian. L_(small hex.) = (1/3) L_H ⇒ Area_(sm. hex) = ((3(√3))/2) un^2 ⇒ Area_(hex. star) = 3 (√3) un^2 Notes: 1• Area_(Hexagon) = 2× (1/2)(2L+L)L cos 30= ((3(√3))/2) L^2 2• Side length_(equilat. triang) = 2×L cos 30= (√3) L 3• Area_(equilat. triang) = (1/2) 3L^2 sin 60= ((3 (√3))/4) L^2