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How is the volume of the pyramid calculated?

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 try to give it a geometric definition. We represent several points (A1, A2, ..., An) on the plane and one more (E) that does not belong to it. So, if the point E (vertex) is connected to the vertices of the polygon formed by the points A1, A2, ..., An (base), we obtain a polyhedron, which is called a pyramid. It is obvious that the vertices of the polygon at the base of the pyramid can be as many as you like, and depending on their number the pyramid can be called triangular and quadrangular, pentagonal, etc.

If you look closely at the pyramid, it becomes clear why it is also defined differently - as a geometric figure that has a polygon at the base, and triangles joined by a common vertex as lateral faces.

Since the pyramid is a spatial figure, it also has such a quantitative characteristic as volume. The volume of the pyramid is calculated according to the well-known formula of volume equal to one third of the product of the base of the pyramid at its height:

The volume of the pyramid in the derivation of the formula is initially calculated for the triangular, taking as a basis a constant relationship connecting this quantity with the volume of a triangular prism having the same base and height, which, as it turns out, is three times that volume.

And since any pyramid is divided into triangular, and its volume does not depend on the constructions performed in the proof, the validity of the reduced volume formula is obvious.

Apart from all the pyramids, there are regular ones, which have a regular polygon at the base . As for the height of the pyramid , it must "end" in the center of the base.

In the case of an irregular polygon at the base, the calculation of the base area requires:

  • Break it into triangles and squares;
  • To calculate the area of each of them;
  • Add the received data.

In the case of a regular polygon at the base of the pyramid, its area is calculated by ready-made formulas, so the volume of the regular pyramid is calculated quite simply.

For example, in order to calculate the volume of a quadrangular pyramid, if it is correct, draw the length of the side of the right quadrangle (square) in the base into a square and multiply the height by 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 the ball inscribed in the pyramid, the area of its full surface;
  • As two-thirds of the product of the distance between two arbitrarily crossed ribs and the area of the parallelogram that form the middle of the remaining four edges.

The volume of the pyramid is calculated simply and in the case when its height coincides with one of the lateral edges, that is, in the case of a rectangular pyramid.

Speaking about the pyramids, we can not ignore the truncated pyramids obtained by the section of the pyramid parallel to the base plane. Their volume is almost equal to the difference in the volumes of the whole pyramid and the cut off vertex.

The first volume of the pyramid, though not quite in its modern form, however equal to 1/3 the volume of the known prism, was found by Democritus. His method of counting Archimedes called "without proof", since Democritus approached the pyramid as a figure made up of infinitely thin, similar plates.

To the question of finding the volume of a pyramid, vector algebra also "resorted", using the coordinates of its vertices for this. The pyramid, constructed on a triple of vectors a, b, c, is one sixth of the module of the mixed product of given vectors.

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