math extra credit?

Question

If two equilateral triangles of area "A" intersect to form a regular hexagon, then what is the area of the hexagon?

Answer 1

I don't think it's intuitively clear that the sides of the hexagon
are s/3 given that a side of an equilateral triangle is s. So I
looked at the resulting hexagon as one triangle with line segments drawn across 2 sides of the triangle parallel to the third side so as to make a smaller equilateral triangle on each of
the three pairs of sides and using each of the vertices of the
larger triangle as a vertex of each of the smaller triangles. Also
I drew them so that their lengths were 1/3 the side of the large
triangle. Each side of the large triangle has three line segments
with the end ones s/3 leaving the mid segment to be s/3. Those
mid segments together with the drawn segments mentioned
earlier make up 6 sides to the hexagon and each side is s/3.
Now the hexagon's area can be computed by drawing a line
from one vertex to an opposite one thus making 2 identical
trapezoids. With vigorous use of Pythagoras, one base of the trapezoid is s/3, the other is 2s/3 and the height is s3^(1/2)/6.
So the area of one trapezoid is (1/2)(2s/3 + s/3)(s3^(1/2)/6)
=(s^2)3^(1/2)/12, so double that for the area of the whole
hexagon and you get (s^2)3^(1/2)/6.
Now the area of the original triangle is A=(1/2)sh. Solving for h
in terms of s is s3^(1/2)/2 and sub'ing back into A we get
A=s^2(3)^(1/2)/4. So s^2=4A/3^(1/2). Now use that for s^2 in the area of the hexagon and we get {4A/3^(1/2)}{3^(1/2)}/6=
(2/3)A. So grandpa is right. The first 3 answers are wrong. There's no crime in being wrong but may I call your attention to the smugness with which septupus asserts his wrong answer.
I suggest he stick to linguistics and leave the math to others.

Answer 2

The answer is that the area of the hexagon is two thirds the area of one triangle:

Area of Hexagon = (2/3)A.

You can do a graphical proof.

Draw one triangle with three equal sides. Draw an identical triangle on top of it but upside down, so you end up with a Star of David shape.
[Picture: http://en.wikipedia.org/wiki/Image:Star_of_David.svg ]
You can see the hexagon shape in the middle. Draw three lines in the hexagon to connect opposite corners.
Now the original triangle is composed of 9 smaller equilateral triangles and the hexagon is made up of 6 of these.
Therefore, Area of Hexagon = (6/9) x (Area of Triangle)
So, Area of Hexagon = (2/3)A.

Answer 3

you will get a regular hexagon with side equalling 1/3rd of side of the equilateral triangleby the intersection of the two equilateral triangles.
since the side of the regular hexagon is a/3 the area of polygon=3√3/2 (a/3)^2 = 3√3/2*(a^2/9 )
Area of euilateral triangle A= (a^2)*(√3/4)
so area of hexagon/ara of equilateral triangle
= [3√3/2*(a^2/9)] / [(a^2)*(√3/4)]
=2/3
i.e area of hexagon = 2/3*A

Answer 4

the only way to make a hexagon is by puting them together

So the total area will be 2*A
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| A A |
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Answer 5

2A