area of a regular pentagon?

Question

if each side of the penagon is 10 cm how do i find the area?

thanks in advance for helping me!

Answer 1

The area of a regular pentagon is approximately 1.72048a², where a is a side. In this case, a = 10:
A = 1.72048(10²) = 1.72048(100) = 172.048 cm²

This is an approximation. To see where I got the formula from, visit http://en.wikipedia.org/wiki/Pentagon

Hope this helps!

Answer 2

The formula for area of a regular polygon is A = 1/2 ap where a = apothem (distance from center to midpoint of a side) and p = perimeter. To get the apothem, draw the pentagon and put in an apothem and a radius (goes from center to a vertex) to one end of the same side. The angle formed by these is 360/10 or 36 degrees, and the side opposite is 10/2 = 5 (the apothem bisects the side). So the apothem can be found by tan(36) = 5/a, about 6.88. Now multiply this by 1/2 then by perimeter (50) to get 172.05.

Sorry for the lecture, probably more than you wanted.

Answer 3

In general, you can break the regular-gon into small triangles, find the area on one and multiply by the number of sides.

The central angle of a pentagon is 360/5 = 72. The other two angles are equal and the three sum to 180. (54). Now use trig to find the height, multiply 1/2 Base x height to find area. It works with any polygon.

Answer 4

Given that any regular polygon can of side n can be reduced to n isosceles triangles with vertex angle = 360°/n and equal sides r,
then Area = n * ½r²sin(360°/n)
= n/2 * r²sin(360°/n)

Also length of each side L = 2*rsin(180°/n).
So r = L/(2sin(180°/n))

Thus Area = n/2 * (L/(2sin(180°/n)))² * sin(360°/n)
= n/2 * L² /(4sin²(180°/n)) * 2sin(180°/n)cos(180°/n)
= ¼ n L² cot(180°/n)

So when n = 5 and L = 10
A = ¼ * 5 * 100 cot 36°
= 125 * cot 36°
≈ 172.05 cm²

Answer 5

The questioner may be asking for the EXACT value using radicals, which can be done for a pentagon. I'm trying to remember how that's done, maybe it'll come back to me.

* * * * *

OK, I cheated and found it on a Web site. The exact value is:

A = s^2 * sqrt(25 + 10sqrt(5)) / 4

Resolving the constant to five-decimal places, you get A = 1.72048s^2.

http://en.wikipedia.org/wiki/Pentagon