Pyramid (geometry)

Source: Wikipedia, the free encyclopedia.

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Definition

Parts of a pyramid

A pyramid is a polyhedron that may be formed by connecting a polygonal base and a point, called the

Heron of Alexandria defined it as the figure by putting the point together with a polygonal base.[5]

A prismatoid is defined as a polyhedron where its vertices lie on two parallel planes, with its lateral faces are triangles, trapezoids, and parallelograms.[6] Pyramids are classified as prismatoid.[7]

Classification and types

The family of a regular polygonal base pyramid: tetrahedron, square pyramid, pentagonal pyramid, and hexagonal pyramid.

A right pyramid is a pyramid where the base is circumscribed about the circle and the altitude of the pyramid meets at the circle's center.

skeleton may be represented as the wheel graph.[17]

Pyramids with rectangular and rhombic bases

A right pyramid may also have a base with an irregular polygon. Examples are the pyramids with rectangle and rhombus as their bases. These two pyramids have the symmetry of C2v of order 4.

A pyramid truncated by an inclined plane
A pentagram-base pyramid.

The type of pyramids can be derived in many ways. The base regularity of a pyramid's base may be classified based on the type of polygon, and one example is the pyramid with

regular star polygon as its base, known as the star pyramid.[18] The pyramid cut off by a plane is called a truncated pyramid; if the truncation plane is parallel to the base of a pyramid, it is called a frustum
.

Mensuration

The surface area is the total area of each polyhedra's faces. In the case of a pyramid, its surface area is the sum of the area of triangles and the area of the polygonal base.

The volume of a pyramid is the one-third product of the base's area and the height. Given that is the base's area and is the height of a pyramid. Mathematically, the volume of a pyramid is:[19]

The volume of a pyramid was recorded back in ancient Egypt, where they calculated the volume of a square frustum, suggesting they acquainted the volume of a square pyramid.[20] The formula of volume for a general pyramid was discovered by Indian mathematician Aryabhata, where he quoted in his Aryabhatiya that the volume of a pyramid is incorrectly the half product of area's base and the height.[21]

Generalization

Main article: Hyperpyramid
4-dimensional hyperpyramid with a cube as base

The hyperpyramid is the generalization of a pyramid in n-dimensional space. In the case of the pyramid, one connects all vertices of the base, a polygon in a plane, to a point outside the plane, which is the

peak. The pyramid's height is the distance of the peak from the plane. This construction gets generalized to n dimensions. The base becomes a (n − 1)-polytope in a (n − 1)-dimensional hyperplane. A point called the apex is located outside the hyperplane and gets connected to all the vertices of the polytope and the distance of the apex from the hyperplane is called height.[22]

The n-dimensional volume of a n-dimensional hyperpyramid can be computed as follows:

Here Vn denotes the n-dimensional volume of the hyperpyramid. A denotes the (n − 1)-dimensional volume of the base and h the height, that is the distance between the apex and the (n − 1)-dimensional hyperplane containing the base A.[22]

References

  1. ^ "Henry George Liddell, Robert Scott, A Greek-English Lexicon, πυραμίς", www.perseus.tufts.edu.
  2. ^ The word meant "a kind of cake of roasted wheat-grains preserved in honey"; the Egyptian pyramids were named after its form. See Beekes, Robert S. (2009), Etymological Dictionary of Greek, Brill, p. 1261.
  3. ^ Cromwell, Peter R. (1997), Polyhedra, Cambridge University Press, p. 13.
  4. ISBN 0-471-25183-6
    .
  5. ^ Heath, Thomas (1908), Euclid: The Thirteen Books of the Elements, vol. 3, Cambridge University Press, p. 268.
  6. ^ Alsina, Claudi; Nelsen, Roger B. (2015), A Mathematical Space Odyssey: Solid Geometry in the 21st Century, Mathematical Association of America, p. 85.
  7. doi:10.1007/PL00009307
    .
  8. ^ Polya, G. (1954), Mathematics and Plausible Reasoning: Induction and analogy in mathematics, Princeton University Press, p. 138.
  9. ^ O'Leary, Michael (2010), Revolutions of Geometry, John Wiley & Sons, p. 10.
  10. ^ Humble, Steve (2016), The Experimenter's A-Z of Mathematics: Math Activities with Computer Support, Taylor & Francis, p. 23.
  11. ISBN 978-1-107-10340-5
    . See Chapter 11: Finite Symmetry Groups, 11.3 Pyramids, Prisms, and Antiprisms.
  12. ISBN 978-0-486-48813-4
    .
  13. Zbl 0132.14603
    . See table III, line 1.
  14. ISBN 978-981-15-4470-5
    .
  15. ^ Kelley, W. Michael (2009), The Humongous Book of Geometry Problems, Penguin Group, p. 455.
  16. ISBN 978-3-319-95588-9
  17. ISBN 978-0-8176-8363-4
    .
  18. ISBN 978-0-521-09859-5, archived
    from the original on 2013-12-11
  19. ISBN 978-1-285-19569-8
    .
  20. JSTOR 27957144
    .
  21. ISBN 978-1-4704-7059-3
    .
  22. ^ a b Mathai, A. M. (1999), An Introduction to Geometrical Probability: Distributional Aspects with Applications, Taylor & Francis, p. 42–43.

See also

External links

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