A pyramid is a polyhedron based on a polygon. All faces in turn form triangles that converge at one vertex. Pyramids are triangular, quadrangular and so on. In order to determine which pyramid is in front of you, it is enough to calculate the number of angles at its base. The definition of "pyramid height" is very common in geometry problems in the school curriculum. In the article we will try to consider different ways of finding it.
Parts of the pyramid
Each pyramid consists of the following elements:
- lateral faces that have three angles and converge at the apex;
- apothem is the height that descends from its top;
- the top of the pyramid is a point that connects the side ribs, but does not lie in the plane of the base;
- the base is a polygon on which the vertex does not lie;
- the height of the pyramid is a segment that intersects the top of the pyramid and forms a right angle with its base.
How to find the height of the pyramid, if its volume is known
Through the formula of the volume of the pyramid V = (S * h) / 3 (in the formula V - volume, S - the area of the base, h - the height of the pyramid) we find that h = (3 * V) / S. To consolidate the material, let's immediately solve the problem. In a triangular pyramid, the base area is 50 cm 2 , while its volume is 125 cm 3 . The height of the triangular pyramid, which we need to find, is unknown. Everything is simple here: we insert the data into our formula. We get h = (3 * 125) / 50 = 7.5 cm.
How to find the height of the pyramid, if the length of the diagonal and its edges are known
As we remember, the height of the pyramid forms a right angle with its base. And this means that the height, edge and half of the diagonal together form a right triangle. Many, of course, remember the Pythagorean theorem. Knowing two dimensions, the third value will not be difficult to find. Recall the well-known theorem a² = b² + c², where a is the hypotenuse, and in our case the edge of the pyramid; b - the first leg or half of the diagonal and c - respectively, the second leg, or the height of the pyramid. From this formula, c² = a² - b².
Now the problem: in the correct pyramid, the diagonal is 20 cm, when the length of the rib is 30 cm. It is necessary to find the height. We decide: c² = 30² - 20² = 900-400 = 500. Hence, c = √ 500 = about 22.4.
How to find the height of a truncated pyramid
It is a polygon that has a section parallel to its base. The height of a truncated pyramid is a segment that connects its two bases. The height can be found at the regular pyramid if the lengths of the diagonals of both bases and the edge of the pyramid are known. Suppose that the diagonal of the larger base is d1, while the diagonal of the smaller base is d2, and the edge has a length of - l. To find the height, you can lower the heights at its base from the two upper opposite points of the chart. We see that we have obtained two rectangular triangles, it remains to find the lengths of their legs. To do this, subtract the smaller from the larger diagonal and divide by 2. So we find one leg: a = (d1-d2) / 2. Then, according to the Pythagorean theorem, we can only find the second leg, which is the height of the pyramid.
Now consider this whole thing in practice. We have a task before us. The truncated pyramid has a square at the base, the diagonal length of the larger base is 10 cm, while the smaller one is 6 cm, and the edge is 4 cm. It is required to find the height. To begin with, we find one leg: a = (10-6) / 2 = 2 cm. One leg is 2 cm, and the hypotenuse is 4 cm. It turns out that the second leg or height will be 16-4 = 12, that is, h = √12 = about 3.5 cm.