Question 1014348: A hexagon is cut from a square of side 6 inches.
a. What is the apothem of the hexagon?
b. What is the area of the hexagon?
c. Find the perimeter of the hexagon.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! What kind of a hexagon? It must be a regular hexagon, because that is the kind of hexagon that has an apothem.
How is the regular hexagon expected to be cut out of that square? With two sides parallel to sides of the square,
like this  ?
I assume that is the basis of the answer expected.
That hexagon is inscribed in the circle inscribed in the square.
It is made of six equilateral triangles whose side lengths are half the length of the square side.
 .
The apothem is the height of one of those triangles:  .
According to the Pythagorean theorem,  .
So,  ,
 ,
 , and
  (rounded).
Since the area of a regular polygon can be calculated as
 ,
and the perimeter of a hexagon with  sides is  ,
NOTE:
A more creative person could cut out a slightly larger hexagon,
like this 
The side of that hexagon is about  times larger, and so is the apothem.
The area of that hexagon is about  times larger.
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