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  # 3d geometry
  
  0
  110
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  Rectangle ABCD is the base of pyramid PABCD. If AB = 8, BC = 4, and the height of the pyramid is 6, then what is the volume of PABCD?
  
  Dec 9, 2021
  
  #1
  +203
  0
  
  From https://web2.0calc.com/questions/rectangle-abcd-is-the-base-of-pyramid-pabcd-if-ab
  
  This is a funny lookin pyramid! The formula for the volume of this pyramid is the same as if "P" were above the center of the base:  volume = (1/3)(area of base)(height)
  
  This helped me verify that.
  
  Draw AC.
  
  Now look at triangle ABC.  From the Pythagorean theorem....
  
  AC2   =   22 + 32   =   4 + 9   =   13
  
  AC is in the same plane as AD and AB, so PA is perpendicular to AC.
  
  Look at triangle PAC.  From the Pythagorean theorem again....
  
  AC2 + PA2  =  52
  
  13  +  PA2  =  25
  
  PA2  =  12
  
  PA  =  √12
  
  PA  =  2√3
  
  And....
  
  volume of pyramid  = (1/3)(area of base)(height)
  
  =  (1/3)(  2 * 3  )( PA )
  
  =  (1/3)(6)(2√3)
  
  =  4√3     cubic units
  
  I just remembered how my math teacher explained that the volume of an oblique cylinder is the same as the volume of a right cylinder (with the same height and base).
  
  Imagine a stack of pennies. If you line the stack straight up, it is a right cylinder. And if you lean the stack over, it is oblique. But the height, area of the base, and the volume stays the same.
  
  And in this case, imagine a stack of square pieces of cardboard that get smaller as you go higher. You can stack the squares up to form a right pyramid, or you can slide them to look like the pyramid in this problem. The volume stays the same.
  
  Dec 10, 2021
  edited by AlgebraGuru  Dec 10, 2021
explanation: The document contains a math question about the volume of a pyramid with a rectangular base, and it also includes a detailed answer to the question. The answer involves using the Pythagorean theorem and the formula for the volume of a pyramid.
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  # 3d geometry
  
  0
  110
  1
  
  Rectangle ABCD is the base of pyramid PABCD. If AB = 8, BC = 4, and the height of the pyramid is 6, then what is the volume of PABCD?
  
  Dec 9, 2021
  
  #1
  +203
  0
  
  From https://web2.0calc.com/questions/rectangle-abcd-is-the-base-of-pyramid-pabcd-if-ab
  
  This is a funny lookin pyramid! The formula for the volume of this pyramid is the same as if "P" were above the center of the base:  volume = (1/3)(area of base)(height)
  
  This helped me verify that.
  
  Draw AC.
  
  Now look at triangle ABC.  From the Pythagorean theorem....
  
  AC2   =   22 + 32   =   4 + 9   =   13
  
  AC is in the same plane as AD and AB, so PA is perpendicular to AC.
  
  Look at triangle PAC.  From the Pythagorean theorem again....
  
  AC2 + PA2  =  52
  
  13  +  PA2  =  25
  
  PA2  =  12
  
  PA  =  √12
  
  PA  =  2√3
  
  And....
  
  volume of pyramid  = (1/3)(area of base)(height)
  
  =  (1/3)(  2 * 3  )( PA )
  
  =  (1/3)(6)(2√3)
  
  =  4√3     cubic units
  
  I just remembered how my math teacher explained that the volume of an oblique cylinder is the same as the volume of a right cylinder (with the same height and base).
  
  Imagine a stack of pennies. If you line the stack straight up, it is a right cylinder. And if you lean the stack over, it is oblique. But the height, area of the base, and the volume stays the same.
  
  And in this case, imagine a stack of square pieces of cardboard that get smaller as you go higher. You can stack the squares up to form a right pyramid, or you can slide them to look like the pyramid in this problem. The volume stays the same.
  
  Dec 10, 2021
  edited by AlgebraGuru  Dec 10, 2021
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question about the volume of a pyramid with a rectangular base, and it also includes a detailed answer to the question. The answer involves using the Pythagorean theorem and the formula for the volume of a pyramid.
  
  Label: question-and-answer