Question

Find the total area of the regular pyramid.
Answer in the form of: (XX(Sqrt) XX + X (Sqrt) X)

Answer 1

https://media.glynlyon.com/g_geo_2012/8/groupi78.gif

Answer 2

Surface area? @AtomicFishbowl

Answer 3

The total surface area, I believe so.

Answer 4

nps^2 all over 4tan(180/n)

Answer 5

What is nps? And if I work out the problem that way will you fall into the form I need it to be in?

Answer 6

I also don't have a scientific calculator.

Answer 7

Is the base a regular hexagon? @AtomicFishbowl

Answer 8

n is the number of sides
p is the number of polygons enclosing the polyhedron
e is the number of edges

Answer 9

Yes, I believe so.

Answer 10

I don't understand what p is.

Answer 11

do you know the final answer?

Answer 12

I have no idea...

Answer 13

Will you tell me how to get the final answer in that form please?

Answer 14

@AtomicFishbowl Notice that that you can divide this hexagon-based pyramid in to 6 triangle based pyramid. Do you see?

Answer 15

yes i will :)

Answer 16

Yes I thought about using the Pythagorean theorem but I don't think I have enough information for that

Answer 17

And even if I did I wouldn't know what to do with it.

Answer 18

OK now. Notice that because that the base is a regular hexagon, each triangle based pyramid is the SAME and each has base that is an equilateral triangle with length 6. Are you with me? @AtomicFishbowl

Answer 19

my answer is 654.72 units squared

Answer 20

@aivantettet26 But it has to be in the form of (XX(Sqrt)XX + X (Sqrt) X)
OHH! @genius12 I didn't put together that it was an equilateral triangle. I'm with you, go on.

Answer 21

i mean that's my answer not really the final answer. im waiting for the answer of sir genius12 :)

Answer 22

Lol I just remembered this is surface area. I was thinking of Volume the whole time rofl...Anyway, I'll do the calculations right now and provide the steps if you need it.

Answer 23

Okay, whichever one is fine! I'd appreciate the answer but if you can explain your steps I'd like that too.

Answer 24

Anyway, notice that since the base is a regular hexagon, we can divide the base in two six equilateral triangle each with side length 6. The height of each this equilateral triangles is
\(\bf \frac{\sqrt{3}}{2}a\) where a = 6 hence the height is \(\bf 3\sqrt{3}\) of each equilateral triangle. Employing the pythagorean theorem, we can find the height of the "outer isosceles triangles" with base 6 and the height of these outer triangles will be:\[\bf (8)^2+(3\sqrt{3})^2=64+27=91 \implies h = \sqrt{91}\]Now we can calculate the area of each of these outer isosceles triangles with base 6 and height \(\bf \sqrt{91}\) with the triangle area formula:\[\bf A=\frac{ 6 \times \sqrt{91} }{ 2 }=3\sqrt{91}\]Now since there 6 of these triangles, the total area of the triangles will be:\[\bf 6 \times 3\sqrt{91}=18\sqrt{91}\]Now we calculate the area of the base which is a hexagon that can be divided once again to those 6 same equilateral triangles with height \(\bf 3\sqrt{3}\). The area of each of these equilateral triangles is given by the same triangle area formula:\[\bf \frac{6 \times 3\sqrt{3}}{2}=9\sqrt{3}\]Now there is 6 of them so the total area of the hexagon becomes:\[\bf 6 \times 9\sqrt{3}=54\sqrt{3}\]Now we add the two final values we got and so the total surface area is:\[\bf 54\sqrt{3}+18\sqrt{91}=265.24 \ units^2\]

Answer 25

@AtomicFishbowl @aivantettet26

Answer 26

very nice !