The spatial geometric shape of a prism is an object of study of stereometry. Its characteristics are considered in grades 10-11 of schools. This article is devoted to the study of the properties of a regular prism using the example of triangular, quadrangular and hexagonal figures.
Prism in stereometry
This figure is a polyhedron or polyhedron, consisting of several parallelograms and two identical polygons of arbitrary type. Both polygons are in parallel planes. They are called the base of the figure. An example of a pentagonal prism made of paper is shown below.
This figure can be obtained in space by taking a pentagon and moving it without rotation to a new plane parallel to the original one.
Prism consists of faces (n + 2), vertices (2 * n) and edges (3 * n). Depending on the type of faces, one speaks of straight, inclined, concave, regular, quadrangular and other types of prisms.
When studying this class of figures in stereometry, much attention is paid to considering the properties of regular prisms.
What is a correct figure?
A correct prism is any prism that is straight and has equilateral and equiangular bases. A direct prism is a spatial figure in which the side surface, instead of parallelograms of a general type, is formed by rectangles or squares. At the correct prism, the geometric centers of both bases lie on one vertical line. An important property of the considered type of figures is that the lengths of the side ribs correspond to the height.
If the prism is straight, but its base is an arbitrary polygon, for example, a rhombus, then such a figure will not be correct. The correct foundation implies the equality of all angles and sides. For example, an equilateral triangle has three angles of 60 o , and for a regular hexagon, these angles are 120 o .
Further in the article we give such properties of a regular prism as surface area, volume and length of diagonals using triangular, quadrangular and hexagonal shapes as examples.
Prism with a triangular base
This prism is shown above in the figure. It is formed by two equilateral triangles and three rectangles. We give the properties and formulas of a regular triangular prism .
Let the side of the base be a, and the height (length of the side rib) be h. Then the surface area of the figure is calculated as follows:
S = √3 / 2 * a 2 + 3 * a * h.
The first term on the right side of the equality corresponds to two base areas, the second term is the area of three rectangles that make up the side surfaces.
To determine the volume of the figure, use the following formula:
V = √3 / 4 * a 2 * h.
This expression is obtained from the general formula for the volume of a prism (product of height by the area of the base).
A triangular prism has one single type of diagonals - these are the diagonals of rectangles. The value of their length is calculated as follows:
d = √ (a 2 + h 2 ).
Quadrangular Prism
Before considering the properties of a regular quadrangular prism, we say that it consists of two identical squares that are connected by four rectangles. For many schoolchildren, this figure is known as a rectangular parallelepiped. If its height coincides with the side of the base, then the parallelepiped degenerates into a cube.
The surface area of the studied symmetrical figure can be calculated by the following simple formula:
S = 2 * a 2 + 4 * a * h = 2 * a * (a + 2 * h).
To calculate the volume, use the following expression:
V = a 2 * h.
When solving problems in geometry, the problem often arises of determining the length of the volume diagonals of a rectangular parallelepiped with a square base. The figure has four volumetric diagonals. Their lengths are the same. To determine these lengths, use the formula:
d = √ (2 * a 2 + h 2 ).
If we replace the parameter h in the formulas for S, V, and d with the value a, then we obtain the corresponding expressions for the cube.
Hexagonal Prism
Unlike previous prisms, the correct hexagonal shape is more complex. This complexity is manifested in the calculation of the base area, as well as in determining the lengths of volumetric diagonals (the figure has two types of diagonals).
We begin to consider the properties of a regular hexagonal prism by determining its surface area. It consists of two regular hexagons and six rectangles. The corresponding formula is:
S = 3 * √3 * a 2 + 6 * a * h.
The area of one hexagon with side a is equal to the area of six equilateral triangles with the same side length.
To determine the volume of the figure, substitute the values of side a and height h in the following equality:
V = 3 * √3 / 2 * a 2 * h.
As noted above, there are two types of volume diagonals in a hexagonal prism. One type of diagonals is formed as a result of connecting the opposite vertices of the bases, another type - as a result of connecting the vertices of the bases through one. The lengths of these diagonals are calculated as follows:
d 1 = √ (4 * a 2 + h 2 );
d 2 = √ (3 * a 2 + h 2 ).
The formulas show that the diagonal of the first type is always longer than the diagonal of the second type (d 1 > d 2 ).