Mathematics, 16.04.2020 03:05 catchonyet

The area of a regular pentagon with a radius of 7 cm is

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Answer from: alfonso2382

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 = $\frac{360}{10}$ = 36°

From the right angle OPQ triangle,

sin ∅ = $\frac{Opp}{Hyp}$

Where, ∅ = 36°

Radius = Hyp = 7 cm

Trigonometric ratio becomes,

sin 36° = $\frac{b}{7}$

0.5878 = $\frac{b}{7}$ (∵ sin 36° = 0.5878 )

b = ( 0.5878 × 7 )

b = 4.115 cm

From the right angle OPQ triangle,

cos ∅ = $\frac{Adj}{Hyp}$

Where, Adj = h

cos 36° = $\frac{h}{7}$

0.809017 = $\frac{h}{7}$

h = ( 0.809017 × 7 )

h = 5.663 cm

Step:2

Area of triangle OPR,

$A = \frac{1}{2} (Height )(Base)$

Where, Height, h = 5.663 cm

Base = b + c = 4.115 + 4.115 = 8.23 cm

Area of OPQ becomes,

A = $\frac{1}{2}$ (8.23)(5.663)

= $\frac{1}{2}$ ( 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.

The area of a regular pentagon with a radius of 7 cm is

Answer from: Quest

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Answer from: Quest

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