The Word “the pyramid” is involuntarily associated with the majestic giants in Egypt, right keeping the rest of the pharaohs. Maybe that's why the pyramid as a geometric shape is unmistakably known to all, even children.
However, let's give it a geometric definition. Imagine on the plane a few points (A1,A2,..., AP) and another (E), not prinadlejish her. Now, if point E (the top) to connect the vertices of the polygon formed by the points A1,A2,..., AP (base), you get a polyhedron, which is called a pyramid. It is obvious that the vertices of the polygon at the base of a pyramid can be any number, and depending on their number, the pyramid can be called the triangular and quadrangular, pentagonal, etc.
If you look closely at the pyramid, it becomes clear, why define it again and differently-as a geometrical figure having a base polygon and the lateral faces-triangles United by a common vertex.
Since the pyramid – spatial figure, and she has such a quantitative characteristic of the volume. The volume of a pyramid is calculated by well known formula of volume equal to one third of the works the base of the pyramid to its height is:
The volume of the pyramid in the derivation of formulas originally calculated for the triangle, based on the constant ratio, connecting this size with the volume of the triangular prism having the same base and height, which, as it turns out, three times this amount.
And because any pyramid is divided into triangular, and its volume depends upon the evidence builds, the validity of the formulas for volume – are obvious.
Alone among all of the pyramids are correct, which lies at the base of a regular polygon. As for the height of the pyramid , it needs “end” in the center of the base.
In the case of an irregular polygon in the basis for calculation of the area of the base you will need:
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 a regular pyramid is calculated quite simply.
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For Example, to calculate the volume of a quadrangular pyramid, if it is correct, erect length of sides of a regular quadrilateral (square) at the base of the square and multiplying by the height of the pyramid, divide the obtained result by three.
The Volume of a pyramid can be calculated using other parameters:
The Volume of a pyramid is calculated just in the case when its height coincides with one of the side edges, that is, in the case of a rectangular pyramid.
Speaking of pyramids, it is impossible to ignore also truncated pyramids, obtained by section of the pyramid parallel to the base plane. Their volume is almost equal to the difference between the volumes of a pyramid and cut off the top.
The First volume of a pyramid, though not quite in its modern form, however, is equal to 1/3 of the volume of the prism, he found Democritus. His method of counting Archimedes called “no evidence”, because Democritus came to the pyramid, as the figure composed of infinitely thin, like plates.
The question of finding the volume of a pyramid “turned” and vector algebra using coordinates of its vertices. The pyramid, built on the three vectors a,b,c, equal to one-sixth from the module of the scalar triple product of the given vectors.
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