To successfully solve problems in geometry, it is necessary to clearly understand the terms that this science uses. For example, such are the “straight line”, “plane”, “polyhedron”, “pyramid” and many others. In this article, we answer the question of what is an apothem.
The dual use of the term "apothem"
In geometry, the meaning of the word "apothema" or "apothema", as it is also called, depends on which object it is applied to. There are two fundamentally different classes of figures in which it is one of their characteristics.
First of all, these are flat polygons. What is an apothem for a polygon? This is the height drawn from the geometric center of the figure to any of its sides.
To make it clearer what is at stake, consider a specific example. Suppose there is a regular hexagon shown in the figure below.
The symbol l denotes the length of its side, the letter a - apothem. For the marked triangle, it is not only a height, but also a bisector and a median. It is easy to show that through side l it can be calculated as follows:
a = √3 / 2 * l
Similarly, an apothem is defined for any n-gon.
Secondly, these are the pyramids. What is an apothem for such a figure? This issue requires a more detailed consideration.
Pyramids and their apothems
First, let us define the pyramid in terms of geometry. This figure is a three-dimensional body formed by one n-gon (base) and n triangles (sides). The latter are connected at one point, which is called the peak. The distance from it to the base is the height of the figure. If it falls on the geometric center of an n-gon, then the pyramid is called a straight line. If, in addition, the n-gon has equal angles and sides, then the figure is called regular. The following is an example of a pyramid.
What is an apothem for such a figure? This is the perpendicular that connects the sides of the n-gon to the top of the figure. Obviously, it represents the height of the triangle, which is the side of the pyramid.
Apothem is convenient to use when solving geometric problems with regular pyramids. The fact is that for them all the side faces are equal to each other isosceles triangles. The latter fact means that all n apothems are equal, therefore, for a regular pyramid, we can talk about one such straight line.
Apothem quadrangular pyramid regular
Perhaps the most illustrative example of this figure will be the famous first wonder of the world - the Cheops pyramid. She is in Egypt.
For any such figure with the correct n-angular base, one can give formulas that allow one to determine its apothem through the length a of the side of the polygon, through the side edge b and height h. Here we write the corresponding formulas for a straight pyramid with a square base. Apothem h b for her will be equal to:
h b = √ (b 2 - a 2/4);
h b = √ (h 2 + a 2/4)
The first of these expressions is valid for any regular pyramid, the second - only for a quadrangular.
We show how these formulas can be used to solve the problem.
Geometric problem
Let a straight pyramid with a square base be given. It is necessary to calculate its base area. The apothem of the pyramid is 16 cm, and its height is 2 times greater than the side of the base.
Every student knows: to find the square area, which is the base of the pyramid in question, you need to know its side a. To find it, we use the following formula for apothem:
h b = √ (h 2 + a 2/4)
The value of apothem is known from the conditions of the problem. Since the height h is two times the length of the side a, this expression can be transformed as follows:
h b = √ ((2 * a) 2 + a 2/4) = a / 2 * √17 =>
a = 2 * h b / √17
The area of the square is equal to the product of its sides. Substituting the resulting expression for a, we have:
S = a 2 = 4/17 * h b 2
It remains to substitute in the formula the value of the apotheme from the conditions of the problem and write the answer: S ≈ 60.2 cm 2 .