A pyramid is a spatial polyhedron, or polyhedron, which is found in geometric problems. The main properties of this figure are its volume and surface area, which are calculated from the knowledge of any two of its linear characteristics. One of these characteristics is the apothem of the pyramid. About it will be discussed in the article.
Pyramid figure
Before giving a definition of the apothem of the pyramid, we will get acquainted with the figure itself. A pyramid is a polyhedron that is formed by one n-coal base and n triangles that make up the side surface of the figure.
Every pyramid has a vertex - the connection point of all triangles. The perpendicular drawn from this vertex to the base is called height. If the height intersects the base at the geometric center, then the figure is called a straight line. A straight pyramid having an equilateral base is called regular. The figure shows a pyramid with a hexagonal base, which is viewed from the side of the face and edge.
Apothem of the correct pyramid
It is also called apothema. It means a perpendicular drawn from the top of the pyramid to the side of the base of the figure. By its definition, this perpendicular corresponds to the height of the triangle, which forms the side face of the pyramid.
Since we are considering a regular pyramid with an n-coal base, then all n apofems for it will be the same, since such are isosceles triangles of the side surface of the figure. Note that identical apothems are a property of a regular pyramid. For a figure of a general type (inclined with an irregular n-gon), all n apofems will be different.
Another property of the correct apothem of the pyramid is that it is simultaneously the height, median and bisector of the corresponding triangle. This means that she divides it into two identical right-angled triangles.
Triangular pyramid and formulas for determining its apothem
In any regular pyramid, important linear characteristics are the length of the side of its base, the side edge b, the height h and the apothem h b . These values are related to each other by the corresponding formulas, which can be obtained if we draw a pyramid and consider the necessary rectangular triangles.
A regular triangular pyramid consists of 4 triangular faces, and one of them (the base) must be equilateral. The rest are isosceles in the general case. The apothem of a triangular pyramid can be determined through other quantities using the following formulas:
h b = √ (b 2 - a 2/4);
h b = √ (a 2/12 + h 2 )
The first of these expressions is true for a pyramid with any correct base. The second expression is characteristic exclusively for a triangular pyramid. It shows that the apothem is always greater than the height of the figure.
The apothem of the pyramid should not be confused with that of a polyhedron. In the latter case, the apothem is called the perpendicular segment drawn to the side of the polyhedron from its center. For example, the apothem of an equilateral triangle is √3 / 6 * a.
The task of calculating the apothem
Let the correct pyramid be given with a triangle at the base. It is necessary to calculate its apothem, if it is known that the area of this triangle is 34 cm 2 , and the pyramid itself consists of 4 identical faces.
In accordance with the condition of the problem, we are dealing with a tetrahedron consisting of equilateral triangles. The formula for the area of one face is:
S = √3 / 4 * a 2
Where do we get the length of the side a:
a = 2 * √ (S / √3)
To determine the apopheme h b, we use the formula containing the side edge b. In this case, its length is equal to the length of the base, we have:
h b = √ (b 2 - a 2/4) = √3 / 2 * a
Substituting the value of a through S, we obtain the final formula:
h b = √3 / 2 * 2 * √ (S / √3) = √ (S * √3)
We got a simple formula in which the apothem of the pyramid depends only on the area of its base. If we substitute the value of S from the conditions of the problem, we get the answer: h b ≈ 7.674 cm.