Wedge (geometry)

Short description: Polyhedron defined by two triangles and three trapezoid faces

Wedge

Faces| 2 triangles,
3 quadrilaterals
Edges| 9
Vertices| 6

In solid geometry, a wedge is a polyhedron defined by two triangles and three trapezoid faces. A wedge has five faces, nine edges, and six vertices.

## Contents

* 1 Properties
* 2 Examples
* 3 References
* 4 External links

## Properties

A wedge is a polyhedron of a rectangular base, with the faces are two isosceles triangles and two trapezoids that meet at the top of an edge.[1]. A prismatoid is defined as a polyhedron where its vertices lie on two parallel planes, with its lateral faces are triangles, trapezoids, and parallelograms;[2] the wedge is an example of prismatoid because of its top edge is parallel to the rectangular base.[3] The volume of a wedge is V=bh(a3+c6), where the base rectangle is a by b, c is the apex edge length parallel to a, and h is the height from the base rectangle to the apex edge.[1]

## Examples

A wedge that is parallel

In some special cases, the wedge is the right prism if all edges connecting triangles are equal in length, and the triangular faces are perpendicular to the rectangular base.[3]

Wedges can be created from decomposition of other polyhedra. For instance, the dodecahedron can be divided into a central cube with 6 wedges covering the cube faces. The orientations of the wedges are such that the triangle and trapezoid faces can connect and form a regular pentagon.

Two obtuse wedges can be formed by bisecting a regular tetrahedron on a plane parallel to two opposite edges.

Special cases
Obtuse wedge as a bisected regular tetrahedron |
A wedge constructed from 8 triangular faces and 2 squares. It can be seen as a tetrahedron augmented by two square pyramids.  |
The regular dodecahedron can be decomposed into a central cube and 6 wedges over the 6 square faces.

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## References

1. ↑ 1.0 1.1 Harris, J. W.; Stocker, H. (1998). ""Wedge". §4.5.2". Handbook of Mathematics and Computational Science. New York: Springer. p. 102. ISBN 978-0-387-94746-4. https://books.google.com/books?id=DnKLkOb_YfIC&pg=PA109.
2. ↑ Alsina, Claudi; Nelsen, Roger B. (2015), A Mathematical Space Odyssey: Solid Geometry in the 21st Century, Mathematical Association of America, p. 85, https://books.google.com/books?id=FEl2CgAAQBAJ&pg=PA85 .
3. ↑ 3.0 3.1 Haul, Wm. S. (1893). Mensuration. Ginn & Company. p. 45. https://archive.org/details/mensuration00hallgoog/page/n57/mode/1up?view=theater&q=wedge.

## External links

* Weisstein, Eric W.. "Wedge". http://mathworld.wolfram.com/Wedge.html.

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Convex polyhedra

Platonic solids (regular)|

* tetrahedron
* cube
* octahedron
* dodecahedron
* icosahedron

Archimedean solids
(semiregular or uniform)

* truncated tetrahedron
* cuboctahedron
* truncated cube
* truncated octahedron
* rhombicuboctahedron
* truncated cuboctahedron
* snub cube
* icosidodecahedron
* truncated dodecahedron
* truncated icosahedron
* rhombicosidodecahedron
* truncated icosidodecahedron
* snub dodecahedron

Catalan solids
(duals of Archimedean)

* triakis tetrahedron
* rhombic dodecahedron
* triakis octahedron
* tetrakis hexahedron
* deltoidal icositetrahedron
* disdyakis dodecahedron
* pentagonal icositetrahedron
* rhombic triacontahedron
* triakis icosahedron
* pentakis dodecahedron
* deltoidal hexecontahedron
* disdyakis triacontahedron
* pentagonal hexecontahedron

Dihedral regular|

* dihedron
* hosohedron

Dihedral uniform|

* prisms
* antiprisms

duals:|

* bipyramids
* trapezohedra

Dihedral others|

* pyramids
* truncated trapezohedra
* gyroelongated bipyramid
* cupola
* bicupola
* pyramidal frusta
* bifrustum
* rotunda
* birotunda
* prismatoid
* scutoid

Degenerate polyhedra are in italics.

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