The area of a regular pentagon is 116.516 squared centimeters, if the pentagon has a radius of 7 cm.
Step-by-step explanation:
The given is,
Radius of pentagon - 7 cm
Step:1
Ref the attachment,
The pentagon contain 10 right angled triangle.
Angle of Right angle triangle = [tex]\frac{360}{10}[/tex] = 36°
From the right angle OPQ triangle,
sin ∅ = [tex]\frac{Opp}{Hyp}[/tex]
Where, ∅ = 36°
Radius = Hyp = 7 cm
Trigonometric ratio becomes,
sin 36° = [tex]\frac{b}{7}[/tex]
0.5878 = [tex]\frac{b}{7}[/tex] (∵ sin 36° = 0.5878 )
b = ( 0.5878 × 7 )
b = 4.115 cm
From the right angle OPQ triangle,
cos ∅ = [tex]\frac{Adj}{Hyp}[/tex]
Where, Adj = h
cos 36° = [tex]\frac{h}{7}[/tex]
0.809017 = [tex]\frac{h}{7}[/tex]
h = ( 0.809017 × 7 )
h = 5.663 cm
Step:2
Area of triangle OPR,
[tex]A = \frac{1}{2} (Height )(Base)[/tex]
Where, Height, h = 5.663 cm
Base = b + c = 4.115 + 4.115 = 8.23 cm
Area of OPQ becomes,
A = [tex]\frac{1}{2}[/tex] (8.23)(5.663)
= [tex]\frac{1}{2}[/tex] ( 46.6065)
A = 23.30324 squared centimeters
Step:3
Pentagon contain 5 triangles,
Area of pentagon = 5 × Area of triangle
= 5 × 23.30324
= 116.516 squared centimeters
Area of pentagon = 116.516 squared centimeters
Result:
The area of a regular pentagon is 116.516 squared centimeters, if the pentagon has a radius of 7 cm.
