What is the area of a regular pentagon when perimetre = 1000m? And how do you work it out?

Answer 1

Pentagon consists of 5 triangles
Angle at centre = 72°
base angles = 54° and 54°
base = 200 m
height = h m
1/2 base = 100 m
tan 54° = h / 100
h = 100 tan 54° = 137.6 m
Area of pentagon = (5) (1/2) (200) (137.6) m ²
Area of pentagon = 68,819 m ²

Answer 2

If the perimeter is 1000 m, there are 5 isosceles triangles you can construct, each with a base of 200 m.

The central angle is 360°, and therefore the top angle of each triangle is 360/5 = 72°. Since the triangles are isosceles, the base angles will be (180 -72)/2 = 54°.

To get the height of each triangle, take

tan 54 = h/(200/2) = 1.37638/100
h = 137.638

Since the height is 137.638, and the base is 200, the area of each triangle is 1/2 h X b = 13763.8

There are 5 triangles, so the area of the pentagon = 5 X 13763.8 = 68819 sq. m

On the other hand, you can take the easy way out. The area of a pentagon is given by the simple formula:

1.721s²

where s = length of 1 side (200 in your case)

Answer 3

Theoretically you have a central point with 5 isosceles triangles arranged evenly around that point. As with any isosceles triangle, two sides are of equal length. The third is 200m in length. Since there are 5 of these triangles, the angles are 72 degrees and then two at 54 degrees each.

To find the area of a single isosceles triangle, divide it down the center (a line from the center of the pentagon to the center of each of the five sides) creating 2 right angle triangles. The 72 degree angle is cut in two, reducing it to 36 degrees; the right angle triangle retains one of the original 54 degree angles and of course now has a 90 degree angle.

The short side is 100m so provided you have a simple calculator, you can figure the area of this triangle. Then just multiply by ten as you have ten of these right angle triangles.

Answer 4

There is a formula for finding the area of a regular n-gon with sides of length a:

A = (n a^2) / (4 tan[pi/n])

(you can derive it by dividing the polygon up into n isosceles triangles and finding their area)

Since the pentagon has a perimeter of 1000, each side has length 200. n = 5 (because it's a pentagon), so

A = (5 x 200^2) / (4 tan[pi/5]) = 6881910 (ish)

Answer 5

Divide the pentagon into five isosceles triangles each with a base of 200m and angle at the apex of 72 deg.

Calculate the area of one triangle and mulitply by five.