1. In a certain regular square pyramid, all of the edges have length \(12\). Find the volume of the pyramid. (No image for this one.)
2. Donatello starts with a marble cube of side length \(10.\) He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length \(s.\) Find \(s.\)
Here's the image for #2:
1) the formula for the volume of a pyramid is length*width*height/3. We know the length and the width but we don't know the height. Since we know this is a regular pyramid, and we know all edges have the length of twelve, we can draw a triangle with the information we know
height^2 + (1/2*length)^2 = k^2(where k is the length from any corner to the tip of the pyramid.) know we plug in the information we know.
height^2 + 36 = 144
height = 6 root 3.
so then we plug in those values in to the formula for a pyramid and get 12*12*6 root 3 /3