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?
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.
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
volume of pyramid = (1/3)(area of base)(height)
= (1/3)( 2 * 3 )( PA )
= 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.