Heptagonal pyramid, how many corner points

How is the volume of the pyramid calculated?

The word "pyramid" is involuntarily associated with the majestic giants in Egypt who faithfully keep the peace of the pharaohs. Perhaps that is why the pyramid is unmistakably recognized as a geometric figure by everyone, including children.

Still, let's try to give it geometrical definition. We represent several points (A1, A2, ..., An) on the plane and one more (E) that does not belong to it. So if point E (vertex) is connected to the corner points of the polygon formed by points A1, A2, ..., An (base), we get a polyhedron called a pyramid. It is obvious that the corner points 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 square, pentagonal, and so on.

If you look closely at the pyramid, thenit becomes clear why it is also defined differently - as a geometric figure that has a polygon at the base and triangles that are connected by a common corner point as side faces.

Since the pyramid is a spatial figure, and it has such a quantitative property as volume. The volume of the pyramid is calculated according to the well-known volume formula, which is one third of the product of the base of the pyramid in its height:

The volume of the pyramid in deriving the formula is first calculated for a triangle, using a constant relationship that connects that size to the volume of a triangular prism of the same base and height, which it turns out to be three times that volume .

And since each pyramid is divided into a triangle and its volume does not depend on the constructions made in the proof, the validity of the reduced volume formula is obvious.

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

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

For example, to calculate the volume of a quadrilateral pyramid, if it's correct, squeeze the length of the side of the right quadrangle (square) in the base and multiply the height of the pyramid by dividing 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 sphere inscribed in the pyramid, the area of ​​its full surface;
  • as two-thirds of the product, the distance between two arbitrarily crossed ribs and the face of the parallelogram that form the center of the remaining four edges.

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

Speaking of pyramids, one cannot ignore truncated pyramids obtained by having a section of the pyramid parallel to the base plane. Their volume is almost equal to the difference in the volumes of the entire pyramid and the truncated vertex.

However, the first volume of the pyramid, although not quite in itsDemocritus, found a modern shape that was 1/3 the volume of the known prism. His method of counting Archimedes has been called "without evidence" since Democritus approached the pyramid as a figure made of infinitely thin, similar plates.

When asked about the volume of the pyramid "rotated" and the vector algebra, which for this purpose uses the coordinates of its corner points. The pyramid, constructed on the triple of the vectors a, b, c, is one sixth of the modulus of the mixed product of the given vectors.

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