Question

A regular pentagon is inscribed inside a circle. The perimeter of the pentagon is 95 units.

A: What is the approximate measure of the apothem of the pentagon?

B: What is the approximate area of the pentagon?

Choose only one answer each for parts A and B.

A: 6.90 A: 13.08 A: 16.16 A: 11.74 B: 328 B: 768 B: 558 B: 621

Answer 1

side length = 95/5 = 19
apothem = side length / 2 * tan(180/5)
apothem = 19 / 2 * tan(36)
apothem = 19 / 2 * 0.72654
apothem = 13.0756737413

area = (5 * side length * apothem) / 2
area = (5* 19 * 13.07567) / 2
area = 621.094325

Answer 2

A:13.08 B:621

Explanation:

We know that.

Perimeter of regular pentagon =5* side of regular pentagon

or Side of regular pentagon =
`(95)/(5)` units ( perimeter =95 unit given)

or Side of regular pentagon =19 units

Now from drown figure.

∠OAM= half of interior angle of regular pentagon

or ∠OAM= half of 108 degree (interior angle of regular pentagon= 108 degree)

or ∠OAM=54 degree.

Again in triangle AOM

AM⊥OM, AM=half of the side of regular pentagon=19/2 units, and tan∠OAM=OM/AM

or OM=tan54*AM (∠OAM=54 degree)

or apothem (OM)= 1.3763819204711---*9.5≈13.08 units.

Area of regular pentagon in the question = 5* area of ΔAOB (see in figure)

or Area of regular pentagon =5*(0.5*AB*OM) square units

or Area of regular pentagon =5*0.5*19*13.0756---- square units≈621 square units

Hence the approximate length of apothem is 13.08 units and the approximate area of regular pentagon is 621 square units