Question

Use the rectangle hexagon with side length 10 meters to fill in the missing information

Answer 1

Answer:

$A_h=150\sqrt{3}\ m^2$

Step-by-step explanation:

Regular Hexagon

For the explanation of the answer, please refer to the image below. Let's analyze the triangle shown inside of the hexagon. It's a right triangle with sides x,y, and z.

We know that x is half the length of the side length of the hexagon. Thus

$x=5 m$

Note that this triangle repeats itself 12 times into the shape of the hexagon. The internal angle of the triangle is one-twelfth of the complete rotation angle, i.e.

$\theta=360/12=30^o$

Now we have $\theta$, the height of the triangle y is easily found by

$\displaystyle tan30^o=\frac{x}{y}$

Solving for y

$\displaystyle y=\frac{x}{tan30^o}=\frac{5}{ \frac{1} {\sqrt{3} }}=5\sqrt{3}$

The value of z can be found by using

$\displaystyle sin30^o=\frac{x}{z}$

$\displaystyle z=\frac{x}{sin30^o}=\frac{5}{\frac{1}{2}}=10$

The area of the triangle is

$\displaystyle A_t=\frac{xy}{2}=\frac{5\cdot 5\sqrt{3}}{2}=\frac{25\sqrt{3}}{2}$

The area of the hexagon is 12 times the area of the triangle, thus

$\displaystyle A_h=12\cdot A_t=12\cdot \frac{25\sqrt{3}}{2}=150\sqrt{3}$

$\boxed{A_h=150\sqrt{3}\ m^2}$