Question

What is the area of a 15-gon with a perimeter of 90 m

Answer 1

Assuming a regular polygon, then s =90/15=6
A 15- gon has exterior angles = 360/15=24, so interior angles of 156
Form a triangle with a radius, apothem and a half- side
Then the 156/2= 78 degrees, with the apothem being opposite, and the halfside being adjacent= 3
so tan 78= a/3 and a= 3tan 78=14.114

Then area = (1/2)Pa= (1/2)(90)(a)=635.13 m^2

Answer 2

Area of the polygon = 635.13 m²

Explanation:

Since area of the polygon = (1/2) × perimeter × apothem

We have to calculate the are of a 15-gon with a perimeter 90 m.

Here parameter is given and we have to calculate the value of apothem.

Since side of a polygon = perimeter / total number of sides = 90/15 = 6

Now if we take an example of pentagon as shown in the figure, number of triangles formed at the center A is equal to number of sides of a polygon.

And in a triangle formed by ABC in any polygon ∠BAC = 360°/ (n) number of sides.

Now we come back to our question ∠BAC = 360/15 = 24

and ∠BAD = 12°(AD is the perpendicular bisector of BC)

Now tan 12 = 3/ h

⇒ 0.2125 = 3/ h

⇒ h = 3/0.2125 = 14.114 m

Now we apply the formula to calculate the area of the polygon.

Area = (1/2) × perimeter × apothem (h)

=(1/2)×90×14.114 = 635.13 m²