The school geometry course is divided into two broad sections: planimetry and stereometry. Stereometry studies spatial figures and their characteristics. In this article we will consider what a direct prism is and give formulas describing its properties such as diagonal lengths, volume and surface area.
What is a prism?
When students are asked to name the definition of a prism, they respond that this figure represents two identical parallel polygons, the sides of which are connected by parallelograms. This definition is as general as possible, since it does not impose conditions on the shape of the polygons, on their mutual arrangement in parallel planes. In addition, it assumes the presence of connecting parallelograms, the class of which also includes square, rhombus and rectangle. Below you can see what a quadrangular prism is.
We see that a prism is a polyhedron (polyhedron), consisting of n + 2 sides, 2 × n vertices and 3 × n edges, where n is the number of sides (vertices) of one of the polygons.
Both polygons are usually called the bases of the figure, the remaining faces are the sides of the prism.
The concept of a direct prism
There are various types of prisms. So, they talk about regular and irregular figures, about triangular, pentagonal and other prisms, there are convex and concave figures, finally, they are inclined and straight. We’ll talk about the latter in more detail.
A direct prism is such a figure of the studied class of polyhedra, all of whose lateral quadrangles have right angles. There are only two types of such quadrangles - a rectangle and a square.
The considered form of the figure has an important property: the height of the straight prism is equal to the length of its lateral edge. Note that all side edges of the figure are equal to each other. As for the side faces, in the general case they are not equal to each other. Their equality is possible if, in addition to the prism being a straight line, it is still correct.
The figure below shows a straight figure with a pentagonal base. It can be seen that all of its side faces are rectangles.
Diagonal prisms and its linear parameters
The main linear characteristics of any prism are its height h and the lengths of the sides of its base a i , where i = 1, ..., n. If the base is a regular polygon, then to describe its properties it is enough to know the length a of one side. Knowing the marked linear parameters allows you to uniquely determine such properties of the figure as its volume or surface.
Diagonals of a direct prism are segments that connect any two non-adjacent vertices. Such diagonals can be of three types:
- lying in the planes of the base;
- located in the planes of the side rectangles;
- volume figures.
The lengths of those diagonals that relate to the base should be determined depending on the type of n-gon.
The diagonals of the side rectangles are calculated by the following formula:
d 1i = √ (a i 2 + h 2 ).
To determine the volume diagonals, it is necessary to know the value of the length of the corresponding diagonal of the base and height. If some base diagonal is denoted by the letter d 0i , then the volume diagonal d 2i is calculated as follows:
d 2i = √ (d 0i 2 + h 2 ).
For example, in the case of a correct quadrangular prism, the length of the volume diagonal will be equal to:
d 2 = √ (2 × a 2 + h 2 ).
Note that a direct triangular prism has only one of the three types of diagonals mentioned above: the diagonal of the side.
The surface of the studied class of figures
A surface area is a collection of areas of all faces of a figure. To visualize all the faces, you should do a scan of the prism. As an example, such a scan for a pentagonal figure is shown below.
We see that the number of planar figures is n + 2, with n being rectangles. To calculate the area of the entire sweep, add up the area of two identical bases and the area of all the rectangles. Then the corresponding formula will look like:
S = 2 × S o + h × ∑ i = 1 n (a i ).
From this equality it can be seen that the lateral surface area for the studied type of prisms is equal to the product of the height of the figure by the perimeter of its base.
The base area S o can be calculated using the appropriate geometric formula. For example, if the base of the direct prism is a right triangle, then we get:
S o = a 1 × a 2/2.
Where a 1 and a 2 are the legs of the triangle.
If the base is an n-gon with equal angles and sides, then the following formula will be fair:
S o = n / 4 × ctg (pi / n) × a 2 .
Volume formula
Determining the volume of a prism of any kind is not a difficult task if the values of its base area S o and height h are known. Multiplying these values among themselves, we get the volume of the V figure, that is:
V = S o × h.
Since the direct prism parameter h is equal to the length of the side edge, the whole problem of calculating the volume is reduced to the calculation of the area S o . Above, we have already said a few words and have given a couple of formulas to determine S o . Here we just note that in the case of a base of arbitrary shape, it should be divided into simple segments (triangles, rectangles), calculate the area of each, and then add all the areas to get S o .