Question

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

Answer 1

The area of a regular pentagon is 116.516 squared centimeters, if the pentagon has a radius of 7 cm.

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 =
`(360)/(10)` = 36°

From the right angle OPQ triangle,

sin ∅ =
`(Opp)/(Hyp)`

Where, ∅ = 36°

Radius = Hyp = 7 cm

Trigonometric ratio becomes,

sin 36° =
`(b)/(7)`

0.5878 =
`(b)/(7)` (∵ sin 36° = 0.5878 )

b = ( 0.5878 × 7 )

b = 4.115 cm

From the right angle OPQ triangle,

cos ∅ =
`(Adj)/(Hyp)`

Where, Adj = h

cos 36° =
`(h)/(7)`

0.809017 =
`(h)/(7)`

h = ( 0.809017 × 7 )

h = 5.663 cm

Step:2

Area of triangle OPR,

`A = (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 =
`(1)/(2)` (8.23)(5.663)

=
`(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.