The pyramid along with the prism is a perfect polyhedron in three-dimensional space, their geometric characteristics are studied in high school. In this article, we consider what pyramids are, what elements they consist of, and briefly describe the correct pyramids.
Geometric shape pyramid
From the point of view of geometry, the pyramid is a spatial figure consisting of one polygon and several triangles. Getting this shape is easy enough. To do this, take a polygon with n sides, then select an arbitrary point in space that will not lie in the plane of the polygon, and connect each vertex of the polygon with this point. Obviously, the figure thus formed will have n triangles connected to each other at the same vertex.
To visualize the geometric shape of the described figure, we present the figure.
A quadrangular pyramid is shown here, the base of which is a quadrangle, and the side surface is formed by four triangles having a common vertex.
Pyramid elements
Like any polyhedron, a pyramid is formed by three types of elements:
Facets are parts of planes that separate the internal volume of a figure from the surrounding space. If the pyramid at the base contains an n-gon, then the number of its faces is always n + 1. Of these, n sides are triangular, and one side is the n-carbon base.
Vertices - points where three or more faces of the figure intersect. The base region contains n vertices, each of which is formed by two triangular faces and a base. The point where n triangular sides connect is called the top of the pyramid. Thus, the figure in question consists of n + 1 vertices.
Ribs are straight lines that appear when two faces intersect. Each edge from its ends is bounded by two vertices. Any pyramid with an n-gon at the base contains 2 * n edges. Half of this number, that is, n, is formed solely by the intersection of the side triangles.
Possible types of figures
The name of the figure in question is uniquely determined by the type of polygon at the base. For example, if it has three angles and three sides, then the pyramid will be triangular, if four - quadrangular, and so on.
A polygon can be convex and concave, as well as regular and general type. All this also determines the type of pyramid.
An important point in determining the type of figure is the position of the top of the pyramid relative to its base. The perpendicular segment drawn from the top to the polygonal base is called the height of the figure. If this segment intersects the base in its geometric center (for a triangle it is the intersection of medians, for a quadrangle it is the intersection of diagonals), then the figure is called a straight line. Otherwise, they talk about an inclined pyramid.
If the n-gon of the base is correct (an equilateral triangle, square and others), and the figure is a straight line, then it is called a regular pyramid.
The figure above shows several pyramids that differ in the number of sides of the polygon at the base.
Properties of Regular Pyramids
These pyramids from other figures of this class are distinguished by a high degree of symmetry. In this regard, it is convenient to carry out various geometric calculations with them, for example, volume or surface area.
The correct pyramid contains an n-gon at the base, the area of which is uniquely determined from the knowledge of the length of its side. The lateral surface of the figure is formed by n identical triangles that are equilateral. The edges of the regular pyramid, located on the side surface, are equal to each other. The value of the length of this rib is often used in calculating the apothemes of the figure and determining the surface area.
The height of the regular pyramid is the second important characteristic of the figure (the first is the length of the edge of the base). Height is used when calculating volume.
Any plane parallel to the base that intersects the side faces of the pyramid leads to the formation of a polygonal section. It is homothetic with respect to the base polygon. The described slice operation leads to the formation of a whole class of new figures - truncated regular pyramids.
The most famous pyramids
Of course, these are the regular quadrangular pyramids of the Egyptian pharaohs. In a place called Giza there are more than 100 of these stone monuments, the perfection of the design and the accuracy of the geometric parameters of which to this day continue to amaze scientists. The largest of them is the Cheops pyramid, whose height is about 146 meters, and the side length is about 230 meters.
For what exactly did these pyramids serve, as well as with the help of which mechanisms and when they were built, no one knows to this day.