# Quadrangular Prism: Formula And Volume, Characteristics

A **quadrangular prism** is one whose surface is formed by two equal bases that are quadrilaterals and by four lateral faces that are parallelograms. They can be classified according to their angle of inclination, as well as the shape of their base.

A prism is an irregular geometric body that has flat faces and these enclose a finite volume, which is based on two polygons and lateral faces that are parallelograms. Depending on the number of sides of the polygons of the bases, the prisms can be: triangular, quadrangular, pentagonal, among others.

**Characteristics How many faces, vertices and edges does it have?**

A quadrangular-based prism is a polyhedral figure that has two equal and parallel bases, and four rectangles that are the lateral faces that join the corresponding sides of the two bases.

The quadrangular prism can be differentiated from the other types of prisms, because it has the following elements:

**Bases (B)**

They are two polygons formed by four sides (quadrilateral), which are equal and parallel.

**Faces (C)**

In total, this type of prism has six faces:

- Four side faces formed by rectangles.
- Two faces that are the quadrilaterals that form the bases.

**Vertices (V)**

They are those points where three faces of the prism coincide, in this case there are 8 vertices in total.

**Edges: (A)**

They are segments where two faces of the prism meet and these are:

- Base edges: it is the union line between a lateral face and a base, there are 8 in total.
- Side edges: it is the lateral union line between two faces, there are 4 in total.

The number of edges of a polyhedron can also be calculated using Euler’s theorem, if the number of vertices and faces are known; thus for the quadrangular prism it is calculated as follows:

Number of Edges = Number of faces + number of vertices – 2.

Number of Edges = 6 + 8 – 2.

Number of Edges = 12.

**Height (h)**

The height of the quadrangular prism is measured as the distance between its two bases.

**Classification**

Quadrangular prisms can be classified according to their angle of inclination, which can be straight or oblique:

**Right quadrangular prisms**

They have two equal and parallel faces, which are the bases of the prism, their lateral faces are formed by squares or rectangles, in this way their lateral edges are all equal and their length will be equal to the height of the prism.

The total area is determined by the area and perimeter of its base, by the height of the prism:

At = A _{lateral} + 2A _{base.}

**Oblique quadrangular prisms**

This prism type is characterized in that its side faces form angles oblique dihedral with bases, namely, that its sides are not perpendicular to the base, because these have a degree of inclination may be more or less than 90 ^{or} .

Their lateral faces are generally parallelograms with a rhombus or rhomboid shape, and they may have one or more rectangular faces. Another characteristic of these prisms is that their height is different from the measurement of their lateral edges.

The area of an oblique quadrangular prism is calculated almost the same as the previous ones, adding the area of the bases with the lateral area; the only difference is the way its lateral area is calculated.

The area of the side is calculated with a lateral edge and the perimeter of the cross section of the prism, which is just where an angle formed of 90 ^{or} with each of the side.

A _{total} = 2 _{* }_{Base} area + Perimeter _{sr * }_{Side} edge_{}

The volume of all types of prisms is calculated by multiplying the area of the base by the height:

V = _{Base} area _{*} height = A _{b } _{*} h.

In the same way, quadrangular prisms can be classified according to the type of quadrilateral that the bases form (regular and irregular):

**Regular quadrangular prism**

It is one that has two squares as a base, and its lateral faces are equal rectangles. Its axis is an ideal line that runs parallel to its faces and ends in the center of its two bases.

To determine the total area of a quadrangular prism, the area of its base and the lateral area must be calculated, such that:

At = A _{lateral} + 2A _{base.}

Where:

The lateral area corresponds to the area of a rectangle; that is to say:

_{Side} A = Base _{*} Height = B _{*} h.

The area of the base corresponds to the area of a square:

A _{base} = 2 (Side _{*} Side) = 2L ^{2}

To determine the volume, multiply the area of the base by the height:

V = A _{base } _{*} Height = L ^{2 }_{*} h

**Irregular quadrangular prism**

This type of prism is characterized because its bases are not square; they can have bases consisting of unequal sides, and five cases are presented where:

**to. The bases are rectangular**

Its surface is formed by two rectangular bases and by four lateral faces that are also rectangles, all equal and parallel.

To determine its total area, each area of the six rectangles that form it, two bases, two small lateral faces and the two large lateral faces is calculated:

Area = 2 (a _{*} b + a _{*} h + b _{*} h)

**b. The bases are rhombuses:**

Its surface is formed by two rhombus-shaped bases and by four rectangles that are the lateral faces, to calculate its total area, it must be determined:

- Base area (rhombus) = (major diagonal
_{*}minor diagonal) ÷ 2. - Lateral Area = perimeter of base
_{*}height = 4 (sides of base) * h

Thus, the total area is: A _{T} = A _{lateral} + 2A _{base.}

**c. The bases are rhomboid**

Its surface is formed by two rhomboid-shaped bases, and by four rectangles that are the lateral faces, its total area is given by:

- Base area (rhomboid) = base
_{*}relative height = B * h. - Lateral Area = perimeter of the base
_{*}height = 2 (side a + side b)_{*}h - Thus the total area is: A
_{T}= A_{lateral}+ 2A_{base.}

**d. The bases are trapezoids**

Its surface is formed by two bases in the shape of trapezoids, and by four rectangles that are the lateral faces, its total area is given by:

- Base area (trapezoid) = h
_{*}[(side a + side b) ÷ (2)]. - Lateral area = perimeter of base
_{*}height = (a + b + c + d) * h - Thus the total area is: A
_{T}= A_{lateral}+ 2A_{base.}

**and. The bases are trapezoids**

Its surface is formed by two trapezoid-shaped bases, and by four rectangles that are the lateral faces, its total area is given by:

- Base area (trapezoid) = = (diagonal
_{1 *}diagonal_{2}) ÷ 2. - Lateral Area = perimeter of the base
_{*}height = 2 (side a_{*}side b * h. - Thus the total area is: A
_{T}= A_{lateral}+ 2A_{base.}

In summary, to determine the area of any regular quadrangular prism, it is only necessary to calculate the area of the quadrilateral that is the base, its perimeter and the height that the prism will have, in general, its formula would be:

_{Total} area = 2 _{* }_{Base} area + _{Base} perimeter _{*} height = A = 2A _{b} + P _{b } _{*} h.

To calculate the volume for these types of prisms, the same formula is used which is:

Volume = _{Base} area _{*} height = A _{b } _{*} h.

**References**

- Ángel Ruiz, HB (2006). Geometries. CR Technology,.
- Daniel C. Alexander, GM (2014). Elementary Geometry for College Students. Cengage Learning.
- Maguiña, RM (2011). Geometry Background. Lima: UNMSM Pre-university Center.
- Ortiz Francisco, OF (2017). Mathematics 2.
- Pérez, A. Á. (1998). Álvarez Second Degree Encyclopedia.
- Pugh, A. (1976). Polyhedra: A visual approach. California: Berkeley.
- Rodríguez, FJ (2012). Descriptive geometry. Volume I. Dihedral System. Donostiarra Sa.