word "pyramid" involuntarily associated with the majestic giants in Egypt, right storing rest of the pharaohs.Perhaps that is why the pyramid as a geometrical figure unmistakably know everything, even the children.
Nevertheless, try to give it a geometric definition.Represent several points in the plane (A1, A2, ..., An) and another (E), it is not prinadlezhayshuyu.So, if the point E (top) to connect to the vertices of the polygon formed by the points A1, A2, ..., Ap (base), you 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 can be called a triangular pyramid and a quadrangular, pentagonal, etc.
If you look closely to the pyramid, it becomes clear why it is also determined in another way - as a geometric shape having at the base of a polygon, and as the side faces - triangles, united by a common vertex.
Since pyramid - dimensional figure, then she has such a quantitative characteristic as the volume.The volume of the pyramid is calculated by the well-known formula of volume equal to one third of the work base of the pyramid to its height:
volume of the pyramid in the derivation originally calculated for the triangular, based on constant relation between the this value to the volume of a triangular prism having the same base and heightwhich, as it turns out, more than three times this volume.
And as any pyramid is divided into triangular, and its amount does not depend on proof constructions performed, the validity of this formula of volume - is obvious.
alone among all the pyramids are correct, which lies at the base of a regular polygon.As for the height of the pyramid, it should be "terminated" in the center of the base.
In the case of an irregular polygon in the base for calculating the footprint required:
- divide it into triangles and squares;
- calculate the area of each of them;
- to lay down the data.
In the case of a regular polygon at the base of the pyramid, its area is calculated from the set formula, so the volume of a regular pyramid is calculated quite simply.
example, to calculate the volume of a quadrangular pyramid, if it is correct, the correct length of the side erected quadrangle (square) at the base of the square, and multiplying by the height of the pyramid, the product obtained is divided by three.
volume of the pyramid can be calculated using other parameters:
- a third of the work radius of the sphere inscribed in a pyramid on a total area of its surface;
- two-thirds of the product of the distance between two randomly selected skew edges and the area of a parallelogram which form the middle of the remaining four ribs.
volume of the pyramid is calculated only 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, we can not ignore as truncated pyramids obtained section of the pyramid parallel to the base plane.Their volume is almost equal to the difference between the volume of the entire pyramid and cut off the top.
first volume of the pyramid, though not quite in its present form, however, equal to 1/3 of the volume of the known prism found Democritus.His method of calculation of Archimedes called "no evidence", as Democritus approached the pyramid, as a figure, which is composed of infinitely thin, like plates.
the question of finding the volume of a pyramid "turned" and vector algebra, using the coordinates of its vertices.The pyramid, built on three vectors a, b, c, equal to one-sixth of the module given a mixed product of vectors.
