Wedge (geometry)

Last updated April 28, 2019

Wedge

Faces	2 triangles, 3 quadrilaterals
Edges	9
Vertices	6
Dual polyhedron	trigonal bipyramid
Properties	convex

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.

In mathematics, solid geometry is the traditional name for the geometry of three-dimensional Euclidean space.

Polyhedron solid in three dimensions with flat faces

In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron.

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted $.$

In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively.

Comparisons:

A wedge is a parallelepiped where a face has collapsed into a line.
A quadrilaterally-based pyramid is a wedge in which one of the edges between two trapezoid faces has collapsed into a point.

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square or as a cuboid to a rectangle. In Euclidean geometry, its definition encompasses all four concepts. In this context of affine geometry, in which angles are not differentiated, its definition admits only parallelograms and parallelepipeds. Three equivalent definitions of parallelepiped are

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base. A pyramid with an n-sided base has n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.

Volume

For a rectangle based wedge, the volume 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 the height from the base rectangle to the apex edge.

Apex (geometry) Peak or top of a geometric figure

In geometry, an apex is the vertex which is in some sense the "highest" of the figure to which it belongs. The term is typically used to refer to the vertex opposite from some "base."

Examples

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.

In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

Cube A geometric shape with 6 square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

A triangular prism is a special case wedge with the two triangle faces being translationally congruent.

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

Special cases
Triangular prism (Parallel triangle wedge)	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.

Related Research Articles

An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices.

In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is the only radially equilateral convex polyhedron.

In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

Rhombicuboctahedron Archimedean solid with eight triangular and eighteen square faces

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.

Parallelogram quadrilateral with two pairs of parallel sides

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

Rhombus quadrilateral in which all sides have the same length

In plane Euclidean geometry, a rhombus is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

Trapezoid convex quadrilateral with at least one pair of parallel sides

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid in American and Canadian English but as a trapezium in English outside North America. The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast to the special cases below.

In geometry, a prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids.

In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C₂₄, icositetrachoron,, octaplex (short for "octahedral complex"), icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.

In geometry, a dodecagon or 12-gon is any twelve-sided polygon.

In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of the same measure. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides are parallel, and the two other sides are of equal length. The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure.

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted $ABCD .$

References

Harris, J. W., & Stocker, H. "Wedge". §4.5.2 in Handbook of Mathematics and Computational Science. New York: Springer, p. 102, 1998. ISBN 978-0-387-94746-4

External links

Weisstein, Eric W. "Wedge". MathWorld .

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