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Contents

• 
  1Further terminology
  
• 
  2Measurements and equations
  
  • 
    2.1Volume
    
  • 
    2.2Center of mass
    
  • 
    2.3Right circular cone
    
    • 
      2.3.1Volume
      
    • 
      2.3.2Slant height
      
    • 
      2.3.3Surface area
      
    • 
      2.3.4Circular sector
      
    • 
      2.3.5Equation form
      
  • 
    2.4Elliptic cone
    
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  3Projective geometry
  
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  4Generalizations
  
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  5See also
  
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  6Notes
  
• 
  7References
  
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  8External links
  

Further terminology[edit]
The perimeter of the base of a cone is called the "directrix", and each of the line segments between the directrix and apex is a "generatrix" or "generating line" of the lateral surface. (For the connection between this sense of the term "directrix" and the directrix of a conic section, see Dandelin spheres.)
The "base radius" of a circular cone is the radius of its base; often this is simply called the radius of the cone. The aperture of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle θ to the axis, the aperture is 2θ.

Illustration from Problemata mathematica... published in Acta Eruditorum, 1734
A cone with a region including its apex cut off by a plane is called a "truncated cone"; if the truncation plane is parallel to the cone's base, it is called a frustum.^[1] An "elliptical cone" is a cone with an elliptical base.^[1] A "generalized cone" is the surface created by the set of lines passing through a vertex and every point on a boundary (also see visual hull).
Measurements and equations[edit]
Volume[edit]
The volume {\displaystyle V}  of any conic solid is one third of the product of the area of the base {\displaystyle A_{B}}  and the height {\displaystyle h} ^[4]
{\displaystyle V={\frac {1}{3}}A_{B}h.}
In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral
{\displaystyle \int x^{2}dx={\tfrac {1}{3}}x^{3}}

Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying Cavalieri's principle – specifically, comparing the cone to a (vertically scaled) right square pyramid, which forms one third of a cube. This formula cannot be proven without using such infinitesimal arguments – unlike the 2-dimensional formulae for polyhedral area, though similar to the area of the circle – and hence admitted less rigorous proofs before the advent of calculus, with the ancient Greeks using the method of exhaustion. This is essentially the content of Hilbert's third problem – more precisely, not all polyhedral pyramids are scissors congruent (can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.^[5]
Center of mass[edit]
The center of mass of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.