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Right Circular Cone Parametric Formulae

In common usage in elementary geometry, cones are assumed to be right circular, where right means that the axis passes through the centre of the base at right angles to its plane, and circular means that the base is a circle.

Often the term cone means right cone that follows the simple formula - height h and base radius r oriented along z-axis with vertex pointing up and base at z=0.

Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base. 

In general, however, the base may be any shape and the apex may lie anywhere. But in most cases the base is bounded with the apex lying outside the plane of the base. For instance a pyramid is technically a cone with a polygonal base.

The formulae for calculating three dimensional right circular cones are:

For a circular cone with radius r and height h, the formula for volume becomes

V = \frac{1}{3} \pi r^2 h.

For a right circular cone, the surface area A is

A =\pi r^2 + \pi r s\,  where   s = \sqrt{r^2 + h^2}  is the slant height.

The first term in the area formula, πr2, is the area of the base, while the second term, πrs, is the area of the lateral surface.

A right circular cone with height h and aperture 2θ, whose axis is the z coordinate axis and whose apex is the origin, is described parametrically as

S(s,t,u) = \left(u \tan s \cos t, u \tan s \sin t, u \right)

where s,t,u range over [0,θ), [0,2π), and [0,h], respectively.

 
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