PLEASE PLEASE HELP! NEED HELP WITH GEOMETRY HOMEWORK!

Question

Identify the area of a regular pentagon with side length 15 cm. rounded to the nearest tenth.

Answer 1

Draw straight lines from each corner of the pentagon to the center of the pentagon to create 5 identical triangles. The length of the base b of each of these triangles is 15cm, because each is a side of the pentagon. If you know the height h of these triangles, which is the length from the center of the pentagon to the midpoint of a side and is called the apothem, you can easily calculate the area A of the pentagon:

A = 5 x b x h / 2 = 75cm x h / 2.

How do you calculate the length of the apothem h? You need to calculate the values for some angles in these triangles and use trigonometry. There are 5 isosceles triangles that meet in the center of the pentagon and complete a 360 degree full circuit, so each has a central angle of 360/5 = 72 degrees. Since the angles of a triangle total 180 degrees and these are isosceles triangles the two other angles = 1/2 x (180 - 72) = 54 degrees each. Looking at 1/2 of one of these isosceles triangles, where one side is length h and the side perpendicular to h has length 1/2 x 15 cm = 7.5cm, you can use:

tan(54) = opposite/adjacent = h/7.5cm to find the value of h:

h = 7.5cm x tan(54) = 10.3228644035cm



So, this is the area A of a pentagon with 15cm sides rounded to the nearest tenth:

A = 75cm x 10.3228644035cm / 2 = 387.1 cm²

Answer 2

First determine the length of the apothegm.

7.5 / a = tan 36°

7.5 / tan 36° = a

7.5 / 0.7265 = a

10.32 = a

A = (0.5)(p)(a), where p represents the perimeter

A = (0.5)(75)(10.32)

A = 387

Answer 3

Using the perimeter (P) and the apothem (distance from center to the midpoint of a side) marked in the diagram as "a":

the area is calculated as A = 1/2Pa

To calculate "a" we need to know the angle so we can use a trig ratio. The entire interior angle (θ) of a regular polygon is the sum of the angles divided by the number of sides. The sum of the angles is (n - 2)180 = (5 - 2)180 = 3•180 = 540. One angle is 540/5 = 108°. For the triangle at the bottom of the diagram, the angle is half of the 108° or 54°. So to calculate "a", we can say:

tan(54) = a/7.5 or

a = 7.5tan(54) or

a = 10.32286

The perimeter is 15•5 = 75

A = 1/2(P)(a)

A = 1/2(75)(10.32286)

A = 387.1 sq cm