SOLUTION: Find the area of an equilateral triangle (regular 3-gon) with the given measurement if the apothem is 6inches

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Question 315581: Find the area of an equilateral triangle (regular 3-gon) with the given measurement if the apothem is 6inches


Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The apothem of an equilateral triangle is,
Ap=%28sqrt%283%29%2F6%29%2AS
where S is the length of the side.
The height of the triangle can be found using Pythagorean therorem,
%28S%2F2%29%5E2%2BH%5E2=S%5E2
S%5E2%2F4%2BH%5E2=S%5E2
H%5E2=%283%2F4%29S%5E2
H=%28sqrt%283%29%2F2%29%2AS
The area of the triangle is given by,
A=%281%2F2%29bh=%281%2F2%29S%28sqrt%283%29%2F2%29S=%28sqrt%283%29%2F4%29S%5E2
From the apothem formula,
Ap=%28sqrt%283%29%2F6%29%2AS
S=%286%2Fsqrt%283%29%29Ap
S%5E2=%2836%2F3%29A%5E2=12Ap%5E2
So then
A=%28sqrt%283%29%2F4%29%2812Ap%5E2%29
A=%28sqrt%283%29%2F4%29%2812%2A36%29
A=108sqrt%283%29