I have two solutions for you.
Trevor,
I like the way you have approached this question but I have a problem with the details. A regular hexagon is made up of 6 equilateral triangles
![](trevor1.1.gif)
and each of these triangles has half the perimeter of the hexagon and hence half the perimeter of the larger triangle. But why half the area?
![](trevor1.2.gif)
Penny
Hi Trevor. Here is a solution using some algebra. Let's try figuring out the ratio using perimeter of 12a and see if we get the same ratio as you predicted. I used 12a as it makes some of the expressions come out neatly. Let's draw the triangle:
![](trevor.1.3.gif)
Each side is 4a. We can see that the area of the equilateral triangle is twice the area of a smaller right triangles. The right triangle's area is 1/2 base x height, and the height (b) is obtained through Pythagorus' Theorem from the other two sides. If we put that all together, we get:
![](trevor.1.4.gif)
Now let's look at the hexagon. You are right that it is made up of six equilateral triangles, so each side is 2a this time:
![](trevor.1.5.gif)
Using the same logic as before, let's calculate the area of each small equilateral triangle:
![](trevor.1.6.gif)
We can see by this that the ratio of At is not half of AT, it is 1/4 instead! Now when we look at the ratio of the area of the hexagon to the area of the large triangle, we get:
![](trevor.1.7.gif)
I hope this helps!
Stephen La Rocque>
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