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# Question about Pyramids' Lateral Area and Volume (Geometry)

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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.

Jun 3, 2019

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(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 Jun 3, 2019
edited by hectictar  Jun 4, 2019
#2
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(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

Jun 3, 2019