The pyramid represents a polyhedron in which basis the polygon lies, and sides represent it the triangles possessing the general top. For a regular pyramid the same definition is fair, but in its basis the regular polygon lies. Height of a pyramid is meant as a piece which is carried out from pyramid top to the basis, and this piece is perpendicular to it. It is very easy to find height in a regular pyramid.
It is required to you
- Depending on a situation to know the pyramid volume, the area of side sides of a pyramid, edge length, length of diameter of a polygon in the basis.
Instruction
1. One of ways to find pyramid height, and not only correct is to express it through pyramid volume. The formula by means of which it is possible to learn its volume looks so: V = (S*h)/3 where S is the area of all side sides of a pyramid in the sum, h is height of this pyramid. Then it is possible to bring another out of this formula, for finding of height of a pyramid: h = (3*V)/SK to an example, it is known that the area of side sides of a pyramid of 84 cm², and the volume of a pyramid is equal to 336 cc. Then it is possible to find height so: h = (3*336)/84 = 12 smotvt: height of this pyramid is 12 cm
2. Considering a regular pyramid in which basis the regular polygon lies, it is possible to come to a conclusion that the triangle formed by height, a half of diagonal and one of pyramid sides is a rectangular triangle (for example, it is AEG triangle in the drawing above). According to Pythagorean theorem, the square of a hypotenuse is equal to the sum of squares of legs (a² = b² + with²). In a case with a regular pyramid, the hypotenuse is a side of a pyramid, one of legs - a half of diagonal of a polygon in the basis, and other leg - pyramid height. In that case, knowing length of a side and diagonal, it is possible to calculate also height. As an example it is possible to consider AEG triangle: AE² =²+GA²Отсюда height of a pyramid of GA can be expressed to EG so: GA = √ (AE²-EG²).
3. That it was more clear how to find height of a regular pyramid, it is possible to review an example: in a regular pyramid length of a side is 12 cm, polygon diagonal length in the basis - 8 cm. Proceeding from these data, it is required to find length of height of this pyramid. Decision: 12² = 4² + with², where with - an unknown leg (height) of this pyramid (rectangular triangle).144 = 16 + 128takim in the way, height of this pyramid of √128 or, about, 11.3 cm