Trunk (geometry)

See also: Trunk

A trunk is the part of a solid which is (which “remains”) between two parallel plans. The solid is generally a cone or a Pyramide.

The faces of the solid obtained in the plans of cut are called bases trunk , and the Distance between the two plans of cut is the height trunk .

The Volume of a trunk is the produced its height by the arithmetic Mean of the surface S of its bases and their Mean geometric. Volume V of the trunk is expressed by the formula générale :

$V \ = \ \ frac {H} {3} \ times (B_1 + \ sqrt {B_1 \ times B_2} + B_2)$ ,

where H is the height of the trunk between the two parallel plans, and B₁ and B₂ is the surface S of the bases of the trunk (contained in the parallel plans of cut of the solid. For example, to calculate the volume of a truncated cone orthogonal :

$V \ = \ \ frac {\ pi H} {3} \ times (r_1^2 + r_1 \ times r_2 + r_2^2)$ ,

where H is the distance separating the plans, and r₁ and r₂ is the rays of the two circular bases.

Another average equivalent, easy to memorize, and more intuitive, is to calculate the volume of the cone or the pyramid before cutting the top of it, and cutting off volume from the cone or the pyramid cut at the top. It is then necessary to determine the height h₀ first cone not truncated, and to retain the formulas of computation of the volume of a cone or a pyramid, which are degenerated cases of this preceding one formule :

These cases are obtained for the solids finished while cutting them with a plan only. In this case, the second plan passes by the top of the pyramid or cone and the second base has a surface nulle the formula is simplified en :

$V \ = \ \ frac {h_0 \ times B_1} {3}$ , where H and h₀ is equal.

Other degenerated cases of the principal formula exist with volumes, sometimes called sections because the cut solids are infinite such as the Parallélépipède cylindrical S, trunks or trunks cylindroid, of which all the faces (or right generators) are parallel to same a directing Vecteur, except for two faces by which pass the plans of cut parallèles ; in this case the two parallel bases of cut have equal surfaces, and the formula is simplified en :

$V \ = \ H \ times B_1$ ,

where B₁ is the surface of one of the two parallel bases of cut and H is the distance between the two plans of cut (measured orthogonally in these plans).

The formula can be employed to calculate the volume of any Polyèdre by initially cutting out it in a succession of trunks (or of Tétraèdre S which is pyramidal degenerated trunks, at triangular single base) and by addditionnant their respectifs volumes;: it is enough to choose an unspecified face and to cut the polyhedron with a plan parallel with this face and passing by a top of the polyhedron belonging to no face parallel with the selected face.

See too

Sector
Miter
Cap
Segment

External bonds

http://www.gomath.com/geometry/frustcone.php
http://mathworld.wolfram.com/PyramidalFrustum.html

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