Answer:
The area of the base of the pyramid is 109.2 mm.
Step-by-step explanation:
The area of the base of a hexagonal pyramid is given by the area of a hexagon:
Where:
P: is the perimeter
a: is the apothem
We need to find the perimeter and the apothem.
The perimeter is equal to:
![P = 6*s](https://tex.z-dn.net/?f=%20P%20%3D%206%2As%20)
Where:
s: is the side of the pyramid
And the apothem is:
![a = \frac{\sqrt{3}}{2}*s](https://tex.z-dn.net/?f=%20a%20%3D%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%2As%20)
So, to calculate the apothem and the perimeter we need to calculate the side of the pyramid. We can find it from the volume of the pyramid:
Where:
h: is the height = 4 mm
V: is the volume = 144 mm³
Then, the side is:
![s = \sqrt{\frac{2V}{\sqrt{3}*h}} = \sqrt{\frac{2*144}{\sqrt{3}*4}} = 6.5 mm](https://tex.z-dn.net/?f=%20s%20%3D%20%5Csqrt%7B%5Cfrac%7B2V%7D%7B%5Csqrt%7B3%7D%2Ah%7D%7D%20%3D%20%5Csqrt%7B%5Cfrac%7B2%2A144%7D%7B%5Csqrt%7B3%7D%2A4%7D%7D%20%3D%206.5%20mm%20)
Now, we can find the perimeter and the apothem.
![P = 6*s = 6*6.5 = 39 mm](https://tex.z-dn.net/?f=%20P%20%3D%206%2As%20%3D%206%2A6.5%20%3D%2039%20mm%20)
Finally, the area is:
Therefore, the area of the base of the pyramid is around 109 mm.
I hope it helps you!