Answer
Area of hexagon H:
H = $3x^2\sin\theta$
Work Step by Step
This question can be broken down into two parts. First, finding the area of a triangle A as a section of the overall area of the hexagon H.
The area of a triangle is known to be:
A = $\frac{1}{2}\times b\times h$
Therefore, the area of the hexagon will be:
H = 6 $\times A$
This question asks the final answer H to be expressed in terms of x, and $\sin \theta$.
To do this we must examine the triangle as a section of the overall hexagon H:
Area of triangle = A = $\frac{1}{2}\times b\times h$
b = base = x
h = height;
$\sin\theta$ = $\frac{h}{x}$;
h = $x \times\sin\theta$
Plug in the values of b and h into the equation for A to get:
A = $\frac{1}{2}\times x\times (x\times \sin\theta)$
Which can be simplified to:
A = $\frac{1}{2}\times x^2\times \sin\theta$
A = $\frac{1}{2}x^2\sin\theta$
Now to find H, we plug A into the equation for H.
H = $6 \times A$
H = $6 \times\frac{1}{2}x^2\sin\theta$
H = $3x^2\sin\theta$
