Remembering the majestic buildings of Ancient Egypt, we can call the pyramid the most mysterious geometric figure. In this article, we give the definition of a pyramid. We also consider the pyramid in terms of its constituent elements and its properties.
What is a pyramid?
From the point of view of stereometry (geometry of three-dimensional space), we can give the following definition of a pyramid: a pyramid is a three-dimensional figure, which is limited by triangular faces and one polygonal face. This definition is very simple and geometrically correct. Nevertheless, it does not allow us to clearly imagine what kind of figure we are talking about.
A clearer definition of a pyramid can be given : a pyramid is a figure that results from connecting straight lines of a point in space with the vertices of a selected polygon. Suppose we have an n-gon (n is an integer) located in some plane. We choose an arbitrary point in space that does not lie in the plane of the n-gon. For definiteness, we call this point the letter H. If we connect all the vertices of the n-gon to the point H as segments, then we get a pyramidal surface. If we add an n-coal base to it, then the formed figure will be a pyramid.
Above, the schematic drawing shows a quadrangular pyramid of arbitrary type, constructed according to the method described above.
What does the figure consist of?
Having become acquainted with the definition of a pyramid (a pyramid in geometry is a polyhedron), we can proceed to consider the elements with which it is formed. Let us give an example of a pyramid.
This figure depicts a hexagonal pyramid. This is what it is called because there is a hexagon at the base of the figure. Looking at the figure, we can distinguish three main geometric elements that make up the pyramid:
We begin the characterization of the figure from the tops. They are the "sharp" ends of the pyramid. In total, the figure has n + 1 vertices, which becomes obvious if we recall the n vertices of the base and one point H that does not belong to the base. Point H is a special vertex of the figure, since it is involved in determining the various types of pyramids. The vertex H differs from the other vertices in that all n triangles of the figure intersect in it.
Ribs are segments connecting vertices into a single structure. The edges of the pyramid are of two types: firstly, these are the edges of the base, the number of which is equal to n, secondly, these are the edges of the side surface, their number is also equal to n. Thus, the number of edges of the figure in question is 2 * n.
Facets, or sides, are important elements of any pyramid that give the figure a three-dimensional appearance. The faces of a figure, like edges, are of two types: firstly, it is the base of the pyramid, which is a flat polygon with n vertices and n sides, and secondly, these are triangles of the side surface. The number of triangles of any pyramid is n. As a result, the considered polyhedron consists of n + 1 single faces.
In the middle of the 18th century, a Swiss scientist Leonard Euler published in one of his works a theorem that links the quantities of the considered polyhedron elements into a single equality. The theorem says: if we take the sum of the faces and vertices, and then subtract the number 2 from it, we get the sum of all the edges of the polyhedron. For a pyramid with an n-carbon base, we have:
2 * n = (n + 1) + (n + 1) - 2
Straight and inclined pyramids
When considering what a pyramid is, it was said above that, depending on the number of sides (angles) of a polygonal base, they speak of pyramids of a quadrangular, pentagonal, triangular and so on.
There is another important classification of these figures: they can be straight and inclined. If we draw a perpendicular from the vertex H of the figure to the base, then at some point O it crosses the base of the pyramid. If the point O is the geometric center of the n-gon, then the pyramid will be straight. If O is not this center, then the pyramid will be inclined.
The external difference between these types of figures is visible at a glance. The slanted and straight quadrangular pyramids are shown below.
The pyramids are right
Another important subclass of pyramids is regular shapes. A correct pyramid will be only if the following two geometric conditions are true for it:
- she is direct;
- its base is an equilateral and equilateral n-gon.
What “straight pyramid” means was noted in the previous paragraph. Here we only explain the second condition. A regular polygon can have any number of sides, starting with 3. So, the right are an equilateral triangle, square, hexagon with the same lengths of sides and angles of 120 o . Note that a rhombus with identical sides is not correct, since it does not have all the angles equal to each other.
Above are four regular pyramids with different regular bases.
It is convenient to carry out any mathematical calculations of parameters precisely for the correct figures.
Parameters of the pyramid linear and angular
The linear characteristics of a pyramid of any type make it possible to unambiguously calculate any other properties of the figure. In stereometry, four types of linear parameters are distinguished:
- the length of the side of the base (if the base is incorrect, then it is necessary for the sake of clarity to know the lengths of all sides and the angles between them);
- height;
- the length of the side rib (ribs if the triangles of the side surface are not equal to each other);
- apothem of the figure (the height of the side face drawn from the vertex H).
In addition to these characteristics, in some problems the values of the angles of dihedral between the side face and the base or the angles of linear between the side edge and the base are also given.
In the case of regular pyramids, there are specific mathematical equalities that connect all the marked parameters.
The surface of the pyramid and its area
As you know, consideration of the areas of spatial figures is accepted and convenient to carry out on the examples of their flat scans. To show what is at stake, below is a scan for a regular pentagonal pyramid.
Since all the faces of the pyramid are presented on the plane, it is very simple to calculate their area; for this, the areas of all n triangles and the area of one n-gon should be added. If all the sides and angles of the named figures are known, then it is not difficult to solve the problem by using the formulas for the area of the corresponding figure.
Volume of a geometric figure
Since the pyramid has faces, the volume of space limited by them is the volume of the figure itself. For a pyramid of arbitrary type, the calculation of this value is performed using the following formula:
V = 1/3 * S o * h
That is, if we take the product of height h by the area of the base S o , and then divide the result by three, we get the volume of the figure.
For regular pyramids, V is a function of only two variables: height h and base side a.
The solution to the problem on the example of a triangular pyramid
We fix the knowledge obtained in the article by solving the following problem: it is known that the base of a regular triangular pyramid has a length of 10 cm and the height of the figure is 15 cm. What is the volume and area of this pyramid?
First, we calculate the area of the base, which is an equilateral triangle. The corresponding formula is:
S o = √3 / 4 * a 2 = √3 / 4 * 10 2 = 43.3 cm 2
Now we calculate the apothem of the pyramid. This can be done if we consider a right-angled triangle built on apofem and height. The length of the apotheme h b is calculated as follows:
h b = √ (a 2/12 + h 2 ) = √ (10 2/12 + 15 2 ) = 15.275 cm
Preparatory calculations are carried out, now you can get answers to the problem. So, the surface area of the pyramid will be equal to:
S = S o + 3 * 1/2 * a * h b = 43.3 + 3/2 * 10 * 15.275 = 272.425 cm 2
The volume of the figure is:
V = 1/3 * S o * h = 1/3 * 43.3 * 15 = 216.5 cm 3
In determining the area of the lateral surface, we took into account that all three lateral triangles are the same.