Prism (geometry)

A prism is a solid shape.

It has two flat ends. These ends look the same. The sides connect the two ends.

Prisms have names based on their ends. A prism with five sides is a pentagonal prism.

Some prisms stand straight up. These are called right prisms. A box is a right prism.

Other prisms lean to the side. These are called oblique prisms.

If you slice a prism, the middle looks like the ends. It is a very special shape.

A prism is a solid shape with two flat ends. These ends are called bases. The bases must be the same size and shape. They also must face each other.

The sides of a prism connect the edges of the two bases. These sides are always parallelograms. We name a prism after its base. A prism with a five-sided base is a pentagonal prism.

There are different kinds of prisms. A right prism stands straight up. Its sides are rectangles. A box is a type of right prism.

An oblique prism leans to one side.

You can find many special types. A star prism has star shapes for bases. A twisted prism is made by twisting the top base. This can make a shape called a Schönhardt polyhedron.

You can also make a prism in more than three dimensions. These are called prismatic polytopes. For example, a 4-4 duoprism is a special shape in four dimensions. Prisms can also be used to make a shape called a cylinder. This happens if you add more and more sides to the base.

Imagine you have two identical flat shapes. If you stack them perfectly on top of each other, you create a solid object. This object is called a prism.

A prism is a type of polyhedron, which is a solid with flat faces. It always has two bases that are the same size and shape. These bases are connected by other faces that are all parallelograms. If you slice through a prism anywhere parallel to the bases, the new shape is exactly like the bases. We name these shapes based on their ends. A prism with a five-sided base is a pentagonal prism.

Prisms can stand straight or lean to the side. A right prism stands perfectly upright. Its side faces are rectangles because they meet the bases at a right angle.

A right rectangular prism is often called a cuboid or a simple box. If the bases are squares, we call it a square prism. An oblique prism is different because it leans. In an oblique prism, the side edges are not perpendicular to the bases.

One special oblique prism is called a parallelepiped. This shape has six faces that are all parallelograms.

People have studied these shapes for a very long time. The word prism first appeared in a famous work called "Elements." This work was written by a mathematician named Euclid.

In Book XI of his work, Euclid defined a prism. He described it as a solid figure held by two equal and parallel planes. The other sides must be parallelograms. Some later writers found his definition a bit confusing. They felt it did not explain the bases clearly enough. This caused some debate among geometry students for many years.

There are many ways to change or twist a prism. You can make a star prism using star shapes for bases.

You can also make a twisted prism. This happens when you twist the top base. The simplest twisted prism is called a Schönhardt polyhedron.

If you slice a prism at an angle, you get a truncated prism. A frustum is another shape that looks similar but has different sized bases.

You can even make a crossed prism. This looks like an hourglass because the sides cross each other.

Math lets us see how these shapes grow and change. If you keep adding more sides to a prism's base, it changes. As the number of sides grows toward infinity, the prism looks like a cylinder.

We can also think about prisms in higher dimensions. These are called prismatic polytopes. A 4-dimensional prism is made by moving a 3-dimensional shape into a new direction. For example, a 4-4 duoprism exists in four-dimensional space.

This shows how math can describe shapes we cannot even see.

In geometry, a prism is a specific type of polyhedron. A polyhedron is a solid figure made entirely of flat faces. A prism consists of two identical polygon bases. These bases are parallel to each other. One base is a translated copy of the other, meaning it is moved without being rotated. The remaining faces are called lateral faces. These faces must all be parallelograms that join the corresponding sides of the two bases.

If you take a cross-section parallel to the bases, the resulting shape is always a translation of the bases. This consistency makes prisms a predictable and important part of geometric study.

Prisms are categorized by how they stand in space. A right prism is one where the joining edges and faces are perpendicular to the base faces. In a right prism, all lateral faces are rectangles. A right rectangular prism is often called a cuboid or a rectangular box. If the base is a square, it is called a square prism.

An oblique prism is different because it leans. In an oblique prism, the edges and faces are not perpendicular to the bases. A special case of an oblique prism is the parallelepiped. A parallelepiped is a polyhedron with six parallelogram faces.

We can also classify prisms by the properties of their bases. A regular prism has bases that are regular polygons. A uniform prism, also called a semiregular prism, is a right prism with regular bases where all edges have the same length. In a uniform prism, every lateral face is a square. These shapes are isogonal, meaning they are uniform polyhedra. They belong to one of two infinite series of semiregular polyhedra. The other series is made of antiprisms.

The history of the prism is tied to the foundations of geometry. The term was first used in the "Elements" by the mathematician Euclid. In Book XI, Euclid defined a prism as a solid figure contained by two opposite, equal, and parallel planes, with the rest being parallelograms. However, this definition has faced criticism from later geometry writers. They argued it was not specific enough regarding the nature of the bases. This lack of clarity caused confusion for many generations of mathematicians.

Mathematics provides specific formulas to calculate the measurements of a prism. The volume is the product of the area of the base and the height. The height is the perpendicular distance between the two base faces.

For a right prism, the surface area is calculated by adding twice the base area to the product of the base perimeter and the height. If the base is a regular n-sided polygon with side length $s$, the volume and surface area can be determined using specific trigonometric values. These formulas allow us to quantify the space a prism occupies and the material needed to cover it.

There are many complex variations of the prism shape. A truncated prism is formed when a plane slices through the prism at an angle that is not parallel to the bases. This results in bases that are not congruent. A frustum is a similar construction, but it features trapezoid lateral faces and differently sized top and bottom polygons.

A star prism is a nonconvex polyhedron created using star polygons as bases.

There are even twisted prisms, like the Schönhardt polyhedron, which are constructed by twisting the top base of a uniform prism.

Prisms can also be extended into higher dimensions as prismatic polytopes. An n-dimensional prismatic polytope is created by translating two (n-1)-dimensional polytopes into the next dimension. For example, a 1-polytopic prism is a rectangle made from two translated line segments. A 4-dimensional prism is made from two 3-dimensional polyhedra. One such example in 4D space is the duoprism. A 4-4 duoprism is a lower symmetry form of a tesseract.

This shows how the concept of a prism scales from simple lines to complex, multi-dimensional structures.