Question

A regular square pyramid just fits inside a cube (the base of the pyramid is congruent to a face of the cube and
the height of the pyramid is equal to the height of the cube). A right cone also just fits inside the same cube (the
diameter of the base of the cone, the height of the cone, and the height of the cube are all equal). Which has the
larger volume, the cone or the square pyramid?

Answer 1

Volume of reg sq. pyramid = vp
vp = 1/3×s^2×h, where s = side and h = height
Volume of cone = vc =1/3×h×pi×r^2
Now we know that h is the same for both, and the cones diameter = s of square base, so radius (r) = 1/2 s
so now vc = 1/3×h×pi×(1/2s)^2
let's remove the same items for both vc and vp
so 1/3 and h
now let's plug an arbitrary number into each:
vc = pi (10/2)^2 = 3.14×25 = 78.54
vp = s^2 = 10^2 = 100
So any square pyramid has slightly more volume than the cone