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:
[tex] A_{b} = \frac{P*a}{2} [/tex]
Where:
P: is the perimeter
a: is the apothem
We need to find the perimeter and the apothem.
The perimeter is equal to:
[tex] P = 6*s [/tex]
Where:
s: is the side of the pyramid
And the apothem is:
[tex] a = \frac{\sqrt{3}}{2}*s [/tex]
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:
[tex] V = \frac{\sqrt{3}}{2}*h*s^{2} [/tex]
Where:
h: is the height = 4 mm
V: is the volume = 144 mm³
Then, the side is:
[tex] s = \sqrt{\frac{2V}{\sqrt{3}*h}} = \sqrt{\frac{2*144}{\sqrt{3}*4}} = 6.5 mm [/tex]
Now, we can find the perimeter and the apothem.
[tex] a = \frac{\sqrt{3}}{2}*s = \frac{\sqrt{3}}{2}*6.5 = 5.6 mm [/tex]
[tex] P = 6*s = 6*6.5 = 39 mm [/tex]
Finally, the area is:
[tex] A_{b} = \frac{P*a}{2} = \frac{39*5.6}{2} = 109.2 mm^{2} [/tex]
Therefore, the area of the base of the pyramid is around 109 mm.
I hope it helps you!
