Prism

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The prism is a type of polyhedron formed by two parallel faces that are identical polygons called bases. These figures are joined by the lateral faces that are parallelograms (quadrilaterals whose opposite sides are parallel).

To explain it another way, the prism is a type of polyhedron made up of two bases that are equal. These are joined by the edges, forming the body of the figure.

Let us also remember that a polyhedron is a three-dimensional figure made up of a finite number of faces that are polygons.

Prism elements

The elements of a prism are:

  • Bases: They are two parallel and identical polygons. For example, two squares or two pentagons (as in the figure below).
  • Side faces: They are parallelograms that join the two bases. They can be rectangles, squares, rhombuses, or rhomboids. In the image below, the rectangle ABJF is one of the side faces.
  • Edges: They are the line segments that join the faces of the prism. For example, segment AB in the example below.
  • Vertices: It is the point where three faces of the polyhedron meet, as any of the points A, B, C, D, E, F, G, H, I or J in the prism shown in the lower part.
  • Height: The distance that separates the two bases of the figure. If the prism is straight, the height is equal to the length of the edge of the lateral faces. That is, in the example below, the height measures the same as the edge AJ or BF.
Pentagonal prism

Prism types

Prisms can be classified based on different criteria. First, according to the number of sides of its bases, it can be triangular, quadrangular, pentagonal, hexagonal, etc.

Likewise, they can be regular, when their bases are regular polygons (with equal sides and interior angles to each other), or irregular, when their bases are irregular polygons.

Similarly, they can be straight prisms, when their lateral faces are squares or rectangles, or oblique prisms, when their lateral faces are rhombus or rhomboid.

Right triangular prism Oblique triangular prism

Finally, it is possible to distinguish between convex prisms, when their bases are convex polygons (all interior angles of the faces are less than 180º), and concave prisms, when their bases are concave polygons (at least one interior angle of the base is greater at 180º).

Convex prism Concave prism

Area and volume of a prism

In general, to calculate the area of ​​a prism (Ap) we must multiply the area of ​​the base (Ab) by two and add the lateral area (the sum of the areas of the lateral faces) that we will call AL.

Also, to calculate the volume of a prism, the area of ​​the base is multiplied by the height of the prism (h).

Prism example

Let's look at an example of how to calculate the area and volume of a prism. Suppose it is a straight quadrangular prism where the base is a square whose side is 10 meters. Also, the height of the figure is 12 meters.

First, the area of ​​the base is its side squared, that is, 102 = 100 m2.Meanwhile, to find the lateral area, we must bear in mind that there are four lateral faces, each one is a rectangle with one side measuring 10 meters and the other measuring 12 meters. Therefore, the area of ​​each side face is 10 × 12 = 120 m2 (see rectangle article).

So the lateral area is equal to the area of ​​each lateral face multiplied by 4: 4 × 120 = 480 m2. Then I apply the formula shown above:

Then, we proceed to calculate the volume:

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