solid geometry Solid geometry or stereometry is the geometry of Three-dimensional space, three-dimensional Euclidean space (3D space). A solid figure is the region (mathematics), region of 3D space bounded by a two-dimensional closed surface; for example, a ...

, a wedge is a

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

defined by two

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

s and three

trapezoid In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are ...

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 In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are ...

that meet at the top of an edge.. A

prismatoid In geometry, a prismatoid is a polyhedron whose vertex (geometry), vertices all lie in two parallel Plane (geometry), planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces ...

is defined as a polyhedron where its vertices lie on two parallel planes, with its lateral faces are triangles,

, and parallelograms; the wedge is an example of prismatoid because of its top edge is parallel to the rectangular base. The volume of a wedge is

V = bh \left(\frac+\frac\right),

where the base rectangle is

a

b

c

is the

apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics) A-Bomb Abomination Absorbing Man Abraxas Abyss Abyss is the name of two characters appearing in Ameri ...

edge length parallel to

a

, and

h

is the height from the base rectangle to the apex edge.

Examples

In some special cases, the wedge is the

right prism Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical theor ...

if all edges connecting triangles are equal in length, and the triangular faces are perpendicular to the rectangular base. Wedges can be created from decomposition of other polyhedra. For instance, the

dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...

can be divided into a central

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

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 pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

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

References

External links

* {{Polyhedron navigator Polyhedra Prismatoid polyhedra