Form of polyhedrons, including, and pyramids, have many real objects, for example, the well-known pyramids of Egypt. This geometrical figure has several parameters, height is basic of which.
Instruction
1. Define whether the pyramid which height you need to find under the terms of a task, correct is. That is considered a pyramid at which the basis is any regular polygon (having the equal parties), and height falls in the center of the basis.
2. The first case arises if in pyramids lies basis a square. Carry out height perpendicular to the basis plane. As a result of it, in a pyramid the rectangular triangle will turn out. Its hypotenuse is a pyramid edge, and a bigger leg - its height. The smaller leg of this triangle passes through the diagonal of a square and is in number equal to its half. If the corner between an edge and the plane of the basis of a pyramid and also one of the parties of a square is given, then in this case find pyramid height, using properties of a square and Pythagorean theorem. The leg is equal to a half of diagonal. As the party of a square is equal to a, and at the same time, diagonal is equal to a√2, find a triangle hypotenuse as follows: x=a√2/2cosα
3. Respectively, knowing a hypotenuse and a smaller leg of a triangle, on Pythagorean theorem remove a formula for finding of height of a pyramid: H= √ [(a√2)/2cosα]^2-[(a√2/2) ^2]= √ [a^2/2 * 1-cos^2α/√ cos^2α] =a*tgα/√ 2, where [(1-cos^2α)/cos^2α =tg^2α]
4. If in the basis of a pyramid there is the correct triangle, then its height will form a rectangular triangle with a pyramid edge. The smaller leg passes through basis height. In the correct triangle height at the same time is also a median. From properties of the correct triangle it is known that its smaller leg is equal to a√3/3. Knowing a corner between an edge of a pyramid and the plane of the basis, find a hypotenuse (it is a pyramid edge). Determine height of a pyramid by Pythagorean theorem: H= √ (a√3/3cosα)^2-(a√3/3) ^2=a*tgα/√ 3
5. At some pyramids the basis is to five - or a hexagon. Such pyramid is also considered regular if all parties of its basis are equal. So, for example, you find height of a pentagon as follows: h= √ 5+2√5a/2 where a - the party use pyatiugolnikaety property for finding of an edge of a pyramid, and then and its height. The smaller leg is equal to a half of this height: k= √ 5+2√5a/4
6. Respectively, find a hypotenuse of a rectangular triangle as follows: √ 5+2√5a/4cosαДалее, as well as in the previous cases, height of a pyramid find k/cosα= on Pythagorean theorem: H= √ [(√ 5+2√5a/4cosα)^2-(√ 5+2√5a/4) ^2]