We explain what a quadrangular prism is, its characteristics, faces, vertices, edges, how to calculate the volume, examples and solved exercises

**What is a Square prism?**

He **Square prism** It is a three-dimensional geometric figure, from the family of polyhedrons. It is made up of two equal and parallel faces, in the shape of a quadrilateral, as bases, and four parallelograms on the sides, for a total of six faces.

There are several criteria to classify them, since there are many possibilities for the shape of the faces and the slope. For example, there are the *right quadrangular prisms* and the *inclined quadrangular prisms*.

In the first case, the sides are perpendicular to the base, and then they are rectangles or squares. In the second case, the lateral faces are inclined with respect to the base, therefore, they cannot be rectangles or squares.

Furthermore, the quadrangular prism can be regular or irregular, depending on whether the base is a regular or irregular quadrilateral. The regular quadrilateral is the square, whose four sides and four angles have equal measures.

An example of a special quadrangular prism is the parallelepiped, whose bases are parallelograms. The shapes of the boxes and bricks are inspired by quadrangular prisms, so they are good examples of how to use this geometric figure in practical applications.

**Quadrangular prism characteristics**

Among the most important characteristics of the quadrangular prism, the following stand out:

Their faces are polygon shaped.

It has a total of 6 faces (2 bases and 4 sides), 12 edges or edges, and 8 vertices (corners).

The lateral faces can have the shape of: square, rectangle, parallelogram, rhombus or rhomboid.

Its sides can be straight (they form a 90º angle with the bases) or inclined (there is an angle of less than 90º on the internal side).

The side faces of right prisms can only be squares or rectangles.

The bases of the prism are also called *guidelines*.

If the base is a regular quadrilateral, the quadrangular prism is also regular. Since a plane figure is regular if all its sides have the same measure, the only possibility is that the bases are square.

When the base of the prism is any other quadrilateral than the square, then the prism is considered irregular.

The regular quadrangular prism can be inscribed in a cylinder.

**Quadrangular prism elements**

The five elements of the quadrangular prism are common to all prisms:

**Bases**made up of two identical and parallel quadrilaterals.

**Side faces**are the four parallelograms that border the figure.

**vertices** or corners, common points that have three adjacent sides of the prism.

**edges** or edges, common segment that has two adjacent faces.

**Height**: is the length of a perpendicular segment with endpoints at the bases. When the prism is straight, the height coincides with the measure of the lateral edges.

**straight section, **area of intersection between the prism and a plane that forms 90º with the lateral edges.

The following image shows each of these elements for a right quadrangular prism:

**Faces, vertices and edges**

Of great importance to study the quadrangular prism are the faces, the vertices and the edges:

**faces**

The faces of the prism make a total of 6: the 2 identical bases in the shape of a quadrilateral and the 4 sides or lateral faces in the shape of a parallelogram.

**vertices**

They are the corners of the figure, the point where three adjacent faces meet.

**edges**

They are the intersection segments between the faces of the prism. The edges are classified into:

**base edges**segments in common between the bases and the lateral faces.

**Lateral edges**as its name implies, are the segments in common between the lateral faces.

The figure above shows the two types of edges, marked with arrows of different colors. The number of edges NA can be determined with the *Euler’s theorem* of polyhedra, which relates the number of edges to the number of faces NC and vertices NV:

NA = NC + NV −2

For the quadrangular prism NC = 6 and NV = 8, therefore:

NA = 6 + 8 −2 = 12

Hence the number of edges or edges of the quadrangular prism is 12.

**How to calculate the volume of a quadrangular prism?**

The volume of the prism is understood to be the part of the space enclosed by it, and it is measured in cubic units, which can be cubic meters, cubic centimeters, cubic feet or other appropriate ones, as long as they are cubed in length.

The volume V is always a positive quantity, and in the case of any quadrangular prism, it is given by the product between the area of the base Ab and the height h:

V = Ab × h

**YO) ****Volume of regular quadrangular prism**

Since the bases are square, and the area of the square is its side ℓ squared:

Ab = ℓ2

Then, the volume of the prism whose height is «h» is:

V = ℓ2 × h

**II) ****Volume of the irregular quadrangular prism**

It depends on the shape of the base and the height «h» of the prism:

**1.- Rectangular base prism**

The area of the rectangle with sides “a” and “b” is:

Ab = a × b

So the volume is:

V = a × b × h

**2.- Rhomboid base prism**

The area of the rhombus is the half product of its diagonals «D» and «d»:

And the volume is:

**3.- Rhomboid-shaped base prism**

The area of the rhomboid-shaped base is the product of its base «b» and its relative height «hr» to said base, which is the perpendicular segment that goes from this base to the side parallel to it.

Ab = b × hr

Hence the volume of the prism with this base is:

V= b × hr × h

**4.- Trapezoidal base prism**

Since the area of the trapezoid is half the sum of the parallel sides «a» and «b», multiplied by its height «c»:

The volume of the prism with a trapezoidal base is:

**5.- Trapezoid-shaped base prism**

The area of a symmetric trapezoid is the half product of its diagonals D and d, therefore:

In such a case, the volume of the prism is:

**solved exercise**

A quadrangular prism with a trapezoidal base has a volume of 648 cm3. The parallel sides of the trapezoid measure a = 10 cm and b = 5 cm, while the height of the trapezoid is c = 6 cm. With these data, find the height of the prism.

**Solution**

Given the dimensions of the base, its area can be easily calculated:

And from the formula:

V = Ab × h

Solve for “h”, the height of the prism, since the volume of the prism is known:

H = V/ Ab = 648 cm3 / 45 cm2 = 14.4 cm

**examples**

**rectangular or cuboid prism**

All six faces of this right prism are square or rectangular. The boxes are examples of rectangular prisms, a shape that is also used in numerous objects and constructions such as buildings.

**Cube**

A cube is a regular quadrangular prism, the six faces of which are square-shaped, for example, a dice or the well-known Rubik’s cube game.

The cube is part of the group of Platonic solids, geometric figures that meet two conditions. The first is that each face is a regular polygon and the second is that each vertex has the same number of faces in common.

The cube meets both conditions, since its faces have the shape of a square, which is a regular polygon. And in each of the eight vertices of the cube, three faces of the cube converge.

The remaining Platonic solids are the tetrahedron, the octahedron, the dodecahedron, and the icosahedron.