Wedge (geometry)

Wedge
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.

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 $V=bh\left({\frac {a}{3}}+{\frac {c}{6}}\right),$ 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

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.

References

^ ^a ^b 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.
^ Alsina, Claudi; Nelsen, Roger B. (2015), A Mathematical Space Odyssey: Solid Geometry in the 21st Century, Mathematical Association of America, p. 85, ISBN 978-0-88385-358-0.
^ ^a ^b Haul, Wm. S. (1893). Mensuration. Ginn & Company. p. 45.

External links

Weisstein, Eric W. "Wedge". MathWorld.

[harris-stocker-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.

[alsina-nelsen-2] Alsina, Claudi; Nelsen, Roger B. (2015), A Mathematical Space Odyssey: Solid Geometry in the 21st Century, Mathematical Association of America, p. 85, ISBN 978-0-88385-358-0.

[haul-3] Haul, Wm. S. (1893). Mensuration. Ginn & Company. p. 45.

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