right circular cone

NOUN : The surface generated by a generator passing through a vertex that lies on the perpendicular axis of a circular directrix.

The American Heritage® Dictionary of the English Language: Fourth Edition. 2000.

Whatever the fuck that means.

My 11th grade math analysis definition:
A cone that has a vertex that forms a right angle with the center of it's circular base, and a point on the edge of the circular base.

V = base * height
            3
thusly...

V = pi * r^2 * height
               3

Another cool thing about said cone is it's similarity, useful when doing problems involving different portions of a cone.

    /\
   /  \ --- similar to
  /-..-\
 /      \ ---- this (the whole cone)
/_      _\
  ------

Therefore, to find the volume of the lower part of the cone, you can just find the whole one, and subtract the smaller similar cone. To find the radius of the smaller cone's base, needed to find it's volume, simply use a proportion since you know that the cones are similar.

Using rectangular coordinates, a cone is described as z = ax^2+ay^2, bounded by z = [0, h], (see interval notation) where h is the height of the cone and a is sort of the slope of the cone. Unless of course the cone is that really cool double-cone used to derive the conic sections, which would be notated z = ±(ax^2+ay^2) for any real z.