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Bog'liq
Cone

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

  • 3Projective geometry

  • 4Generalizations

  • 5See also

  • 6Notes

  • 7References

  • 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.

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