Prism (geometry)

In geometry, a prism is a special kind of polyhedron. It has two identical shapes called bases. These bases are usually polygons, like triangles or squares. They are placed parallel to each other. The sides of a prism are flat surfaces called faces. These faces are always parallelograms. Parallelograms are four-sided shapes where opposite sides are equal and parallel.

Prisms are named based on the shape of their bases. For example, a prism with a triangular base is called a triangular prism. One with a pentagonal base is called a pentagonal prism. If you cut the prism with a plane parallel to the bases, the slice will look just like the bases.

The idea of a prism is very old. The word prism comes from the Greek word πρίσμα (prisma), meaning "something sawed." It was first used in Euclid's Elements. This book is very important in the history of mathematics. Euclid described a prism as a solid shape with two parallel, identical faces. The rest of the faces are parallelograms.

Oblique vs right

An oblique prism is a prism where the sides are not straight up and down compared to the base shapes. Picture leaning a box to the side — the sides are slanted. A common example is a parallelepiped, which has six slanted, parallelogram-shaped faces.

A right prism has sides that stand straight up and down, perpendicular to the base shapes. This means all the side faces are rectangles. If you take a shape like a triangle or square and stretch it straight up, you get a right prism. A special kind of right prism with a rectangular base is called a cuboid, which is just a box shape.

Types

A regular prism is a prism with regular polygon bases. This means the shapes on each end are perfectly symmetrical.

A uniform prism is a special type of prism. It is straight (not tilted) and has regular polygon bases with all edges of the same length. Because of this, the side faces of a uniform prism are all squares. All its faces are regular polygons. These prisms are also known as uniform polyhedra and are part of an infinite series of semiregular polyhedra, along with antiprisms. A uniform n-gonal prism can be described using a special mathematical notation called a Schläfli symbol.

Main article: Uniform polyhedra Main articles: Semiregular polyhedra · Antiprisms

Properties

The volume of a prism is found by multiplying the area of its base by its height. This works for all prisms, whether they are right or not. If the base is a regular polygon, there are special ways to calculate the volume using the number of sides, the side length, and the height.

We can also find the surface area of a prism by adding up the areas of all its faces. For a right prism with a regular polygon base, the surface area depends on the base area, the perimeter of the base, and the height of the prism.

Prisms also have interesting symmetry properties. The symmetry of a prism depends on the shape of its base and whether the prism is right or not. Cubes, which are special kinds of prisms, have high symmetry.

Main article: Schlegel diagrams

Similar polytopes

A truncated prism is made when a prism is cut by a plane that is not parallel to its bases. The bases are not the same size, and the sides are not parallelograms.

A twisted prism is a special shape made by twisting the top part of a prism. It cannot be split into simpler shapes without adding new points. The simplest twisted prism has triangle bases.

A frustum is similar to a prism but has trapezoid sides and bases that are different sizes.

A star prism is made from two identical star-shaped bases connected by rectangles. It looks like a regular prism but with star-shaped tops and bottoms.

A crossed prism is made by flipping one base of a prism, turning the sides into crossed rectangles.

A toroidal prism is a special shape without top and bottom bases, made only from rectangular sides. It can only be made with even-sided bases.

A prismatic polytope is a higher-dimensional version of a prism, made by moving a shape into another dimension. For example, a 3D prism made from a polygon has twice as many vertices and edges as the original polygon.