Find the lateral area the regular pyramid. L. A. =

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Find the lateral area the regular pyramid. L. A. =

Geometry
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This is a very awesome tutorial http://www.virtualnerd.com/tutorials/?id=PreAlg_11_01_0031
The lateral area of a regular pyramid is 1/2p*l where p is the perimeter of the base and l is the slant height. The lateral formula for a cone is also 1/2p*l where p is the perimeter (or in this case the circumference) of the base and l is the slant height. It can also be shown as pi*r*l, where r is the radius of the base, or 1/2pi*d*l, where d is the diameter of the radius of the base. However, note that this is only the lateral area formula for a right cone. There can be many formulas for the lateral area of an oblique cone. And what they mean when they say they want the answer in terms of pi is that they do not want you to do any process to it. They want the pi to be in the answer. For example, instead of multiplying 3*pi, the answer would instead be 3pi. Or instead of adding 3 + pi, the answer would actually be 3 + pi.

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If you'd like more help; please let me know. Truly feel that tutorial should shine some light
okay
the base of that pyramid is a hexagon, and a hexagon is made up of 6 equilateral triangles
If we use this formula 1/2p*L We know p = 6*6 We have to find the slant height L
Wouldn't the slant height be 8?
8 is the altitude or height of the pyramid
Okay so how do I find the slant height?
First let's identify the slant height L . I drew it in purple http://prntscr.com/7qqtha
okay
to find L we can construct a right triangle
http://prntscr.com/7qqvbz
okay
we know the leg 8, that is given. the other leg we don't know
correct
but that leg is the altitude for an equilateral triangle as shown here http://prntscr.com/7qqwob
okay
and what do we know about equilateral triangles
an equilateral triangle can be divided into two triangles of 30 60 90 degrees
yes
the ratio of the sides of a 30 60 90 triangle is \( 1 : \sqrt 3 : 2 \)
okay
|dw:1436461601747:dw|
the height of this triangle is the leg we need
okay
so we need 3 : h : 6 to have same ratio as 1 : sqrt(3) : 2
3 : 3 √3 : 6 works
if you divide through by 3, you get 1 : √3 : 2
Is 3 the height?
3√3 is the height
|dw:1436462137479:dw|
So I would do \[\frac{ 1 }{ 2 }(6*6)(3\sqrt{3})\]
almost, now we have to L
oh okay
http://prntscr.com/7qr6c4
We can use pythagorean theorem to find L
8^2 + ( 3 √ 3)^2 = L^2
91?
correct, that would be L^2 take the square root of that
L^2 = 91 L = sqrt(91)
So the lateral area will be $$\Large \frac{ 1 }{ 2 }(6*6)(\sqrt{91})$$
I hope that was clear.
Thank you!

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