+51 Hi! I'm having trouble with 2 math problems from my homework, the theorems are not in the book and I am having trouble finding relavant information online.
(1) Find the lateral area and volume of a pyramid with a rectangular base of dimensions 32 by 66 and with lateral edge 65.
(2) Find the lateral area of a regular pentagonal pyramid with base edge 18 and lateral edge 16.
+9466 (1)
Here, s is a slant height and a is the altitude.
We can find s with the Pythagorean theorem.
s2 + 162 = 652
s2 = 652 - 162
s2 = 3969
s = 63
Now we can find a with the Pythagorean theorem.
a2 + 332 = s2
a2 + 332 = 632
a2 = 632 - 332
a2 = 2880
a = √[ 2880 ]
a = 24√5
*****edit*****
My original answer for lateral area wasn't right because I didn't take into account that the faces aren't all the same.
The same way we found s , we can find the length of the other slant height.
other slant height = √[ 652 - 332 ] = 56
lateral area = 2 * (1/2) * 32 * 63 + 2 * (1/2) * 66 * 56 = 32 * 63 + 66 * 56 = 5712 square units
And we can check this answer with this handy dandy calculator: here
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volume = (1/3) * area of base * altitude
volume = (1/3) * (32 * 66) * a
volume = (1/3) * (32 * 66) * 24√5
volume = 16896√5 cubic units
If you have a question about where any of these numbers came from please ask 
+9466 (2) Here is a sketch of a single face of the regular pentagonal pyramid:
Remember there are 5 faces total because a pentagon has 5 sides. So....
lateral area = 5 * area of triangular face
lateral area = 5 * (1/2)(18)( s )
We can find s with the Pythagorean theorem:
92 + s2 = 162
s2 = 162 - 92
s2 = 175
s = √[ 175 ]
s = 5√7
lateral area = 5 * (1/2)(18)( 5√7 )
lateral area = 225√7 square units
