Prism is one of the famous figures studied in the course of stereometry in secondary schools. In order to be able to calculate various characteristics for figures of this class, it is necessary to know what types of prisms exist. Let's consider this question in more detail.
Prism in stereometry
First of all, we give a definition to the mentioned class of figures. A prism is any polyhedron consisting of two parallel polygonal bases that are interconnected by parallelograms.
You can get this figure like this: you should select an arbitrary polygon on the plane, and then move it to the length of any vector that does not belong to the original plane of the polygon. During such a parallel movement, the sides of the polygon will describe the lateral faces of the future prism, and the final position of the polygon will become the second base of the figure. By the described method, an arbitrary type of prism can be obtained. The figure below shows a triangular prism.
What are the prisms of species?
This is a classification of figures of the class in question. In the General case, this classification is carried out taking into account the features of the polygonal base and the sides of the figure. The following three types of prisms are usually distinguished:
- Straight and oblique (oblique).
- Right and wrong.
- Convex and concave.
The prism of any of these types of classification can have a quadrangular, pentagonal, ..., n-angular base. As for the types of triangular prism, it can be classified only by the first two named points. The triangular prism is always convex.
Below we consider in more detail each of these types of classification and give some useful formulas for calculating the geometric properties of a prism (surface area, volume).
Straight and oblique figures
You can distinguish a direct prism from a slanting prism at a glance. We give the corresponding figure.
Two prisms are shown here (hexagonal on the left and pentagonal on the right). Everyone will say with certainty that the hexagonal is straight and the pentagonal is sloping. What geometric feature distinguishes these prisms? Of course, the type of side face.
In a direct prism, regardless of its base, all faces are rectangles. They can be equal to each other, but can differ, it is only important that they are rectangles, and their dihedral angles with bases are 90 o .
Regarding the inclined figure, it should be said that all or some of the side faces of it are parallelograms, which form indirect dihedral angles with the base .
For all types of direct prisms, the height is the length of the side rib, for inclined figures, the height is always less than their side ribs. Knowing the height of the prism is important when calculating its surface area and volume. For example, the volume formula has the form:
V = S o * h
Where h is the height, S o is the area of one base.
Prisms are right and wrong
Any prism is wrong if it is not straight, or its base is not correct. The issue of direct and inclined prisms was discussed above. Here we consider what the expression "regular polygonal base" means.
A polygon is correct if all its sides are equal (we denote their length by the letter a), and all its angles are also equal to each other. Examples of regular polygons are an equilateral triangle, a square, a hexagon with six angles of 120 o and so on. The area of any regular n-gon is calculated using the following formula:
S n = n / 4 * a 2 * ctg (pi / n)
The diagram below shows how regular prisms look with triangular, square, ..., octagonal bases.
Using the above formula for V, you can write the corresponding expression for the correct figures:
V = n / 4 * a 2 * ctg (pi / n) * h
As for the area of the full surface, for regular prisms it is formed by the areas of two identical bases and n identical rectangles with sides h and a. These facts allow us to write the formula for the surface area of any regular prism:
S = n / 2 * a 2 * ctg (pi / n) + n * a * h
Here, the first term corresponds to the area of two bases, the second term determines the area exclusively of the lateral surface.
Of all the types of prisms that are correct only for a quadrangular, there are proper names. So, a regular quadrilateral prism, for which a ≠ h, is called a rectangular parallelepiped. If this figure has a = h, then they talk about a cube.
Concave shapes
So far, we have considered exclusively convex types of prisms. It is they who are given the main attention when studying the class of figures in question. However, there are also concave prisms. They differ from convex ones in that their bases are concave polygons, starting with a quadrangle.
For example, the figure shows two concave prisms that are made of paper. The left one in the form of a five-pointed star is a decagonal prism, the right one in the form of a six-pointed star is called a twelve-pointed concave direct prism.