Question 967418: The perimeter of a 11-gon is 16.5 meters. Is this enough information to find the area?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you can find informationj about polygons and names of polygons here.
http://www.mathsisfun.com/geometry/polygons.html
an eleven sided polygon is called an hendecagon.
that comes from the greek word for eleven.
the perimeter is 16.5 meters.
divide that by 11 and you get 1.5 meters per side.
each side forms the base of an isosceles triangle with the vertex of that isosceles triangle at the center of the regular hendecagon.
the vertex angle is equal to 360 / 11 = 32.727272...... degrees.
this means that each of the base angles of the isosceles triangle is equal to (180 - 32.727272727272.....(/2 = 73.636363...... degrees.
dro an altitude from the vertex of the isosceles triangle to the base and you form 2 right triangles.
each right triangle has one leg equal to .75 meters (half of the 1.5 meters of the side of the isosceles triangle, and an angle of 73.636363..... degrees.
from that information, you can find the other leg of the right triangle which is the alritude of the isosceles triangle.
that is derived from tan(73.636363....) = opposite / adjacent which becomes:
tan(73.636363...... = altitude / .75
solve for altitude to get altitude = .75 * tan(73.636363.....) which makes the alritude equal to 2.554265429.
now that you have the altitude of the isosceles, which is the same as the height of the isosceles triangle, and you have the base of the isosceles triangle, you can find the area of the isosceles triangle.
the area is equal to 1/2 * base * height = 1/2 * 1.5 * 2.554265429.
this results in the area of the isosceles triangle being equal to 1.915699072 square meters.
multiply that by 11 and you have the area of the hendecagon.
the area of the hendecagon is equal to 11 * 1.915699072 which is equal to 1.07268979 square meters.
fyi, hendecagon is also called undecagon.
the names are inter-changeable.
here's a nice little calculator that does all the calculations for you.
http://www.calculatorsoup.com/calculators/geometry-plane/polygon.php
i do recommend just using it to check your work, since you will have to do the work yourself on a test, so it's good to know how to do it.
the concept here was to find the area of one of the isosceles triangle and then multiply that triangle by 11 to get the area of the hendecagon / undecagon.
each isosceles triangle is formed from lines starting at the center of the hendecagon and terminating at the corner points on the circumference of the hendecagon.
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