The word "pyramid" is involuntarily associated with the majestic giants in Egypt, faithfully keeping the peace of the pharaohs. Maybe that’s why the pyramid as a geometric figure is unmistakably recognized by everyone, even children.
Nevertheless, we will try to give it a geometric definition. We represent on the plane several points (A1, A2, ..., An) and one more (E) that does not belong to it. So, if the point E (vertex) is connected with the vertices of the polygon formed by the points A1, A2, ..., An (base), we get a polyhedron, which is called a pyramid. Obviously, the vertices of the polygon at the base of the pyramid can be any number, and depending on their number, the pyramid can be called triangular and quadrangular, pentagonal, etc.
If you look closely at the pyramid, it will become clear why it is also defined differently - as a geometric figure having a polygon at the base, and triangles united by a common vertex as side faces.
Since the pyramid is a spatial figure, then it also has such a quantitative characteristic as volume. The volume of the pyramid is calculated by the well-known volume formula equal to one third of the product of the base of the pyramid by its height:
When deriving the formula, the volume of the pyramid is initially calculated for a triangular one, taking as a basis a constant relation connecting this quantity with the volume of a triangular prism having the same base and height, which, as it turns out, is three times this volume.
And since any pyramid is divided into triangular ones, and its volume does not depend on the constructions performed in the proof, the validity of the given volume formula is obvious.
Apart from all the pyramids are the regular ones, in which the base is a regular polygon. As for the height of the pyramid , it should "end" in the center of the base.
In the case of an irregular polygon in the base, to calculate the area of the base you will need:
- break it into triangles and squares;
- calculate the area of each of them;
In the case of a regular polygon at the base of the pyramid, its area is calculated using ready-made formulas, so the volume of a regular pyramid is calculated very simply.
For example, to calculate the volume of a quadrangular pyramid, if it is correct, erect the side length of the regular quadrangle (square) at the base into a square and, multiplying by the height of the pyramid, divide the resulting product by three.
The volume of the pyramid can be calculated using other parameters:
- as a third of the product of the radius of a ball inscribed in a pyramid by the area of its full surface;
- as two-thirds of the product of the distance between two arbitrarily taken intersecting edges and the area of the parallelogram, which is formed by the midpoints of the remaining four edges.
The volume of the pyramid is calculated simply in the case when its height coincides with one of the side edges, that is, in the case of a rectangular pyramid.
Speaking of the pyramids, one cannot ignore the truncated pyramids obtained by the section of the pyramid parallel to the base of the plane. Their volume is almost equal to the difference in the volumes of the whole pyramid and the cut off top.
The first volume of the pyramid, though not entirely in its modern form, however, equal to 1/3 of the volume of the prism known to us, was found by Democritus. Archimedes called his method of counting “without proof”, since Democritus approached the pyramid as a figure composed of infinitely thin, similar plates.
Vector algebra also “addressed” the question of finding the volume of the pyramid, using the coordinates of its vertices for this. A pyramid built on a triple of vectors a, b, c is equal to one sixth of the module of the mixed product of given vectors.