The height of the pyramid. How to find it?

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 corners at its base. The definition of "pyramid height" is very often encountered in geometry problems in the school curriculum. In this article, let's try to consider different ways of finding it.

Parts of the pyramid

Each pyramid consists of the following elements:

• Side faces that have three corners and converge at the apex;
• Apophema is the height that descends from its apex;
• The top of the pyramid is the point that connects the side edges, 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 a pyramid, if its volume is known

Through the volume formula of the pyramid V = (S * h) / 3 (in the formula V is the volume, S is the area of the base, and h is the height of the pyramid), we find that h = (3 * V) / S. To fix the material, let's solve the problem immediately. In the triangular pyramid, the area of the base is 50 cm ² , while its volume is 125 cm ³ . The height of the triangular pyramid is unknown, and we need to find it. Here everything is simple: we paste the data into our formula. We obtain h = (3 * 125) / 50 = 7.5 cm.

How to find the height of a pyramid if the length of the diagonal and its edges is known

As we recall, the height of the pyramid forms a right angle with its base. And this means that the height, the edge and half of the diagonal together form a rectangular triangle. Many, of course, remember the theorem of Pythagoras. Knowing the 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 cathetus or half the diagonal and c - respectively, the second cathetus, 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 both the length of the rib is 30 cm. It is necessary to find the height. Solve: 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 the truncated pyramid is a segment that connects its two bases. The height can be found in the correct pyramid if the lengths of the diagonals of both bases are known, as well as the edge of the pyramid. Suppose that the diagonal of the larger base is d1, while the diagonal of the smaller base is d2, and the edge has length-l. To find the height, it is possible to lower the heights on its base from the two upper opposite points of the diagram. We see that we have turned out two rectangular triangles, it remains to find the lengths of their legs. To do this, from the larger diagonal, subtract the smaller and divide by 2. So we find one cut: a = (d1-d2) / 2. After that, according to Pythagoras' theorem, it remains for us to find the second leg, which is the height of the pyramid.

Now let's look at this whole thing in practice. Before us is the task. The truncated pyramid has a square in the base, the length of the diagonal of the larger base is 10 cm, while the smaller one - 6 cm, and the edge equals 4 cm. It is required to find the height. First, we find one cathet: a = (10-6) / 2 = 2 cm. One cathet 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.