For question 5 on page 242(Angles, Triangle and Polygons), could you explain why the triangle was put into 60,30,90 right triangle and how you came up with b/2 *square root of 3 as the height?
Sure. The question is: “The area of a regular hexagon is . What is its perimeter?”
The solution notes that a regular hexagon is made up of six equilateral triangles, and goes from there. I think you’re stuck on this figure from the solution?
![](https://i0.wp.com/pwntestprep.com/wp-content/uploads/2020/08/image.png?resize=213%2C214&ssl=1)
Let’s break it down.
We’re working from the given that the larger triangle is equilateral, which explains why all the sides have length . We know equilateral triangles have three
angles.
![](https://i0.wp.com/pwntestprep.com/wp-content/uploads/2020/08/image-1.png?resize=220%2C227&ssl=1)
Therefore, when we drop a vertical line from the top, we create a right triangle with one of its acute angles being .
![](https://i0.wp.com/pwntestprep.com/wp-content/uploads/2020/08/image-2.png?resize=220%2C227&ssl=1)
If one angle is and another angle is a right angle, then the third angle must be
.
![](https://i0.wp.com/pwntestprep.com/wp-content/uploads/2020/08/image-3.png?resize=220%2C227&ssl=1)
So we know we have a 30-60-90 and we know its hypotenuse has length .
![](https://i0.wp.com/pwntestprep.com/wp-content/uploads/2020/08/image-5.png?resize=144%2C193&ssl=1)
In a 30-60-90, we know the sides are generally, from smallest to largest, ,
, and
. The hypotenuse of
we have here corresponds to
, so the short leg must be half that,
. From there, we know the long leg is
times the short leg, or
.
![](https://i0.wp.com/pwntestprep.com/wp-content/uploads/2020/08/image-4.png?resize=182%2C261&ssl=1)
Does that help?