Question

10 POINTS! HELP!!

Find the area of a regular pentagon with apothem length of about 10.3 meters. Round to the nearest tenth if necessary.

a. 77.3 units^2

b. 386.3 units^2

c. 154.5 units^2

d. 772.5 units^2

Answer 1

b. 386.3 unit^2.

Explanation:

The apotherm is a line drawn from the center of the pentagon to the midpoint of one of the sides.

The area is 5 times the area of one of the triangles (drawn from the center to 2 adjacent vertices of the pentagon) whose height = the apotherm. This triangle is an isosceles triangle with base angles = 108 /2 = 54 degrees, (because the internal angles of a regular pentagon = 108 degrees).

Half of the base of the triangle is obtained using trigonometry:

tan 54 = height / 1/2 base

1/2 base = 10.3 / tan 54

= 7.483 m.

Area of the triangle = 1/2 base * height = 7.483 * 10.3 m

Area of the pentagon = 5 * 7.483 * 10.3

= 385.3 m^2

Answer 2

b. 386.3 unit²

Explanation:

∠AOB = 360 /5 = 72

∠BOS = 72 /2 = 36

tan ∠BOS = tan 36° = (1/2 s) / 10.3 = 0.73

S = 2 x 10.3 x 0.73 = 15.04

Area = (15.04 x 10.3 x 5) / 2 = 387.28 ≈ 386.3