Carcass wrote:
Attachment:
GRE Hexagon ABCDEF is a regular hexagon with a perimeter of 48.jpg
Hexagon ABCDEF is a regular hexagon with a perimeter of 48. What is the volume of the uniform prism shown?
A. \(96\)
B. \(96 \sqrt{3}\)
C. \(1,104\)
D. \(1,472 \sqrt{3}\)
E. \(2,208 \sqrt{3}\)
What we need to find is the area of the hexagonal face, and then multiply that by the height to get the volume.To find the area of a hexagon, you can draw diagonals inside it to create 6 smaller triangles.
It's helpful to know the formula for the sum of angles of an n-sided polygon:
\((n-2)*180\)
In this case, plugging in \(n = 6\) gives 720. Since its mentioned that this is a regular hexagon, each angle is the same. So when we created these diagonals to form the smaller triangles we actually split the 120 degree angles into 60 degree angles.
Adding up all the angles in each individual triangle, you'll notice they are all 60 degrees, so we have 6 equilateral triangles.Regular hexagons also have all equal lengths. So the perimeter of 48 can be divided by 6 to obtain the side length of 8 for each side of the hexagon. And since 8 is one of the sides of the equilateral triangle, all of its sides are 8.
If we find the area of one of the triangles we can find the area of the hexagon by multiplying the area by 6, since we have 6 identical equilateral triangles forming it.Choosing one of the triangles, drop a height from the center of the hexagon to the midpoint of the base of the chosen triangle, splitting the base into two equal sides of length 4 and creating two right triangles.
Using the equilateral triangle ratio for right triangles:
\(base:height:hypotenuse\)
\(x:x\sqrt{3}:2x\)
We find that the height = \(4\sqrt{3}\).
Now, that we have the height and the base of one of the six triangles, we can find its area:
\(8*4\sqrt{3} * \frac{1}{2}\) =
\(16\sqrt{3}\)Multiplying this by 6 we obtain:
\(16\sqrt{3} * 6 = 96\sqrt{3}\)
So the area of the hexagon is \(96\sqrt{3}\)Now, we need to multiply the area of the hexagon by the height of 23 to get the volume of the prism:
\(96\sqrt{3} * 23 = 2,208 \sqrt{3}\)
The final answer is \(2,208 \sqrt{3}\), which is answer choice E