Pyramid - a polyhedron, which lies at the base of the polygon.All faces in turn form triangles that meet at a vertex.The pyramids are triangular, quadrangular and so on.In order to determine what the pyramid in front of you, it is enough to count the number of angles at its base.The definition of "the height of the pyramid" is very common in problems of geometry in the school curriculum.This article will try to consider different ways of finding it.
the pyramid
Each pyramid consists of the following elements:
- side faces that are at three corners and converge at the apex;
- apothem a height that is lowered from its top;
- vertex of the pyramid - a point that connects the lateral edges, but this does not lie in the plane of the base;
- base - a polygon, which is not a vertex;
- height of the pyramid is a segment that crosses the top of the pyramid and forms with its base a right angle.
How to find the height of the pyramid, if you know its volume
A formula for the volume of the pyramid V = (S * h) / 3 (in the formula V - volume, S - area of the base, h - the height of the pyramid)find that h = (3 * V) / S.To consolidate the material, let's solve the problem immediately.The triangular pyramid base area is 50 cm2, while its volume is 125 cm3.Unknown height of a triangular pyramid, and which we need to find.It's simple: insert data into our formula.Get the h = (3 * 125) / 50 = 7.5 cm.
How to find the height of the pyramid, if we know the length of the diagonal ribs and her
As we remember, the height of the pyramid forms with its base a right angle.This means that the height and half the diagonal rib together form a right triangle.Many, of course, remember the Pythagorean theorem.Knowing the two measurements, the third value will be easy to find.Let us recall the well-known theorem a² = b² + c², where a - the hypotenuse, and in this case the edge of the pyramid;b - the first leg or half diagonally and - respectively, the second leg, or the height of the pyramid.This formula c² = a² - b².
Now the problem: in the right diagonal of the pyramid is 20 cm, while the length of the edge - 30 cm. It is necessary to find the height.Solve: c² = 30² - 20² = 900-400 = 500. Hence a = √ 500 = about 22.4.
How to find the height of the truncated pyramid
It is a polygon which has a cross section parallel to the ground.The height of the truncated pyramid - a segment that connects two of its founding.Height can be found in regular pyramid, will be known if the diagonal length of both bases and an edge of the pyramid.Let diagonal greater base equal to d1, while the smaller base diagonal - d2, and the rib has a length - l.To find the height, you can top two opposite points of the diagram height lowered its base.We see what we've got two right triangles, it remains to find the length of the legs.To do this, we subtract the lower most diagonal and divide by 2. So we will find one leg: a = (d1-d2) / 2.Then by the Pythagorean theorem, we can only find the second leg, which is the height of the pyramid.
Now look at all the case in practice.We have before us the task.The truncated pyramid has a square at the base, the larger base diagonal length is 10 cm, while the smaller - 6 cm, and the edge is equal to 4 cm. The height required to find.To find the beginning of one leg of a = (10-6) / 2 = 2 cm. One leg is 2 cm, and the hypotenuse - 4 cm. It turns out that the second leg or height is equal to 16-4 = 12, that is, h =√12 = about 3.5 cm.
