Wedge (geometry)

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

where the base rectangle is ${\displaystyle a}$ by ${\displaystyle b}$, ${\displaystyle c}$ is the apex edge length parallel to ${\displaystyle a}$, and ${\displaystyle h}$ 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

External links

• Weisstein, Eric W. "Wedge". MathWorld.