Английская Википедия:Frustum

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In geometry, a Шаблон:Langnf;Шаблон:Efn (Шаблон:Plural form: frusta or frustums) is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting the solid. In the case of a pyramid, the base faces are polygonal and the side faces are trapezoidal. A right frustum is a right pyramid or a right cone truncated perpendicularly to its axis;[1] otherwise, it is an oblique frustum. In a truncated cone or truncated pyramid, the truncation plane is Шаблон:Em necessarily parallel to the cone's base, as in a frustum. If all its edges are forced to become of the same length, then a frustum becomes a prism (possibly oblique or/and with irregular bases).

Elements, special cases, and related concepts

Файл:Square frustum.png
Square frustum
Файл:Triangulated monorectified tetrahedron.png
A regular octahedron can be augmented on 3 faces to create a triangular frustum

A frustum's axis is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise.

The height of a frustum is the perpendicular distance between the planes of the two bases.

Cones and pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the apex (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of prismatoids.

Two frusta with two congruent bases joined at these congruent bases make a bifrustum.

Formulas

Volume

The formula for the volume of a pyramidal square frustum was introduced by the ancient Egyptian mathematics in what is called the Moscow Mathematical Papyrus, written in the 13th dynasty (Шаблон:Circa):

<math>V = \frac{h}{3}\left(a^2 + ab + b^2\right),</math>

where Шаблон:Mvar and Шаблон:Mvar are the base and top side lengths, and Шаблон:Mvar is the height.

The Egyptians knew the correct formula for the volume of such a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus.

The volume of a conical or pyramidal frustum is the volume of the solid before slicing its "apex" off, minus the volume of this "apex":

<math>V = \frac{h_1 B_1 - h_2 B_2}{3},</math>

where Шаблон:Math and Шаблон:Math are the base and top areas, and Шаблон:Math and Шаблон:Math are the perpendicular heights from the apex to the base and top planes.

Considering that

<math>\frac{B_1}{h_1^2} = \frac{B_2}{h_2^2} = \frac{\sqrt{B_1B_2}}{h_1h_2} = \alpha,</math>

the formula for the volume can be expressed as the third of the product of this proportionality, <math>\alpha</math>, and of the difference of the cubes of the heights Шаблон:Math and Шаблон:Math only:

<math>V = \frac{h_1 \alpha h_1^2 - h_2 \alpha h_2^2}{3} = \alpha\frac{h_1^3 - h_2^3}{3}.</math>

By using the identity Шаблон:Math, one gets:

<math>V = (h_1 - h_2)\alpha\frac{h_1^2 + h_1h_2 + h_2^2}{3},</math>

where Шаблон:Math is the height of the frustum.

Distributing <math>\alpha</math> and substituting from its definition, the Heronian mean of areas Шаблон:Math and Шаблон:Math is obtained:

<math>\frac{B_1 + \sqrt{B_1B_2} + B_2}{3};</math>

the alternative formula is therefore:

<math>V = \frac{h}{3}\left(B_1 + \sqrt{B_1B_2} + B_2\right).</math>

Heron of Alexandria is noted for deriving this formula, and with it, encountering the imaginary unit: the square root of negative one.[2]

In particular:

  • The volume of a circular cone frustum is:
<math>V = \frac{\pi h}{3}\left(r_1^2 + r_1r_2 + r_2^2\right),</math>
where Шаблон:Math and Шаблон:Math are the base and top radii.
<math>V = \frac{nh}{12}\left(a_1^2 + a_1a_2 + a_2^2\right)\cot\frac{\pi}{n},</math>
where Шаблон:Math and Шаблон:Math are the base and top side lengths.
Pyramidal frustum
Pyramidal frustum

Surface area

Файл:CroppedCone.svg
Conical frustum
Файл:Tronco cono 3D.stl
3D model of a conical frustum.

For a right circular conical frustum[3][4]

<math>\begin{align}\text{Lateral surface area}&=\pi\left(r_1+r_2\right)s\\

&=\pi\left(r_1+r_2\right)\sqrt{\left(r_1-r_2\right)^2+h^2}\end{align}</math> and

<math>\begin{align}\text{Total surface area}&=\pi\left(\left(r_1+r_2\right)s+r_1^2+r_2^2\right)\\

&=\pi\left(\left(r_1+r_2\right)\sqrt{\left(r_1-r_2\right)^2+h^2}+r_1^2+r_2^2\right)\end{align}</math> where r1 and r2 are the base and top radii respectively, and s is the slant height of the frustum.

The surface area of a right frustum whose bases are similar regular n-sided polygons is

<math>A= \frac{n}{4}\left[\left(a_1^2+a_2^2\right)\cot \frac{\pi}{n} + \sqrt{\left(a_1^2-a_2^2\right)^2\cot^2 \frac{\pi}{n}+4 h^2\left(a_1+a_2\right)^2} \right]</math>

where a1 and a2 are the sides of the two bases.

Examples

Файл:Rolo-Candies-US.jpg
Rolo brand chocolates approximate a right circular conic frustum, although not flat on top.

See also

Notes

Шаблон:Notelist

References

Шаблон:Reflist

External links

Шаблон:Wiktionary Шаблон:Commons category

Шаблон:Polyhedron navigator Шаблон:Authority control

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  1. ↑ Шаблон:Cite book
  2. ↑ Nahin, Paul. An Imaginary Tale: The story of Шаблон:Sqrt. Princeton University Press. 1998
  3. ↑ Шаблон:Cite web
  4. ↑ Шаблон:Cite journal
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