Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 3$, $BC = 2$, $\overline{PA}\perp \overline{AD}$, $\overline{PA}\perp \overline{AB}$, and $PC = 5$, then what is the volume of $PABCD$?

michaelcai
Oct 10, 2017

#1**+2 **

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....

AC^{2} = 2^{2} + 3^{2} = 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....

AC^{2} + PA^{2} = 5^{2}

13 + PA^{2} = 25

PA^{2} = 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

*edit*

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.

hectictar
Oct 10, 2017

#1**+2 **

Best Answer

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....

AC^{2} = 2^{2} + 3^{2} = 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....

AC^{2} + PA^{2} = 5^{2}

13 + PA^{2} = 25

PA^{2} = 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

*edit*

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.

hectictar
Oct 10, 2017