+0

# 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
The text passage contains a 3D geometry problem asking for the volume of a pyramid with a rectangular base, and it provides a detailed solution to the problem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
+0

# 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
user
The text passage contains a 3D geometry problem asking for the volume of a pyramid with a rectangular base, and it provides a detailed solution to the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER