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Wouldn't a circle-based pyramid look a lot like a cone ? If so, you could probably use the formula for the volume of a cone and get away with it.
Cone: (1/3) * Pi * Radius² * Height Pyramid: (1/3) * (Base Area) * Height
The volume of the pyramid and cone is one third the volume of the corresponding (ie same [size] base and height) prism and cylinder.
Because the formula is the same: volume = 1/3base areaheight
Start with a regular tetrahedron (triangular based pyramid). As you increase the number of sides in the base of the pyramid, the shape becomes more and more like a right cone. In the limit, the base tends to a polygon with an infinite number of sides - a circle, and the pyramid tends to a right cone.