A pyramid is a polyhedron, at the base of whichlies the 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 we will try to consider different ways of finding it.
Parts of the pyramid
Each pyramid consists of the following elements:
How to find the height of a pyramid, if its volume is known
Through the volume formula of the pyramid V = (S * h) / 3 (inV 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 cm2, while its volume is 125 cm3. 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 with itsby the base right angle. 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 leg or half of the diagonal and with - 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 as the length of the rib - 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 thathas 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, by the theorem of Pythagoras, we only need 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 is 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.
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