A hexagonal pyramid has a volume of 144 cubic millimeters and a height of 4 millimeters. What is the area of the base of the pyramid? (Hint: Label the parts you know. Draw a picture if necessary)
This is confusing T^T

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!

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