A cube has a volume of 216 cm^3. A corner is cut off. What is the volume of the pyramid, and what fractional part of the cube does it represent?
Let the cut-off corner contain the vertex (0,0,0).
Find the equation of the plane through the points (s,s,s) where s<3 so only a single corner is cut off. That is x+y+z-s = 0
Now recall that the height of the pyramid formed is the distance from (0,0,0) to that plane: h = s/√3
So the base of the pyramid is an equilateral triangle of side s, with area s^2/4 √3
The volume of the pyramid is thus 1/3 Bh = 1/12 s^2