Pyramid is a spatial polyhedron, or polyhedron, which occurs 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. It will be discussed in the article.
Pyramid shape
Before giving the definition of the apothem of the pyramid, let's get acquainted with the figure itself. The pyramid is a polyhedron, which is formed by one n-gonal 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 the height. If the height intersects the base in the geometric center, then the figure is called a straight line. A straight pyramid with an equilateral base is called a regular pyramid. The figure shows a pyramid with a hexagonal base, which is viewed from the side of the face and edge.
Apothem of the right pyramid
She is also called apotema. It is understood as a perpendicular drawn from the top of the pyramid to the side of the base of the figure. By definition, this perpendicular corresponds to the height of the triangle that forms the side face of the pyramid.
Since we are considering a regular pyramid with an n-gonal base, then all n apothems for it will be the same, since such are the isosceles triangles of the lateral surface of the figure. Note that identical apothems are a property of a regular pyramid. For a figure of a general type (oblique with an irregular n-gon), all n apothems will be different.
Another property of a regular pyramid apothem is that it is simultaneously the height, median and bisector of the corresponding triangle. This means that she divides it into two identical right triangles.
Triangular pyramid and formulas for determining its apothem
In any regular pyramid, the important linear characteristics are the length of the side of its base, the side edge b, the height h and the apothem hb. These quantities are related to each other by the corresponding formulas, which can be obtained by drawing a pyramid and considering the necessary right 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. apothemtriangular pyramid can be determined in terms of other quantities using the following formulas:
hb=√(b2- a2/4);
hb=√(a2/12 + h2)
The first of these expressions is valid for a pyramid with any correct base. The second expression is characteristic only for a triangular pyramid. It shows that the apothem is always greater than the height of the figure.
Don't confuse the apothem of a pyramid with that of a polyhedron. In the latter case, the apothem is a perpendicular segment drawn to the side of the polyhedron from its center. For example, the apothem of an equilateral triangle is √3/6a.
Apothem task
Let a regular pyramid with a triangle at the base be given. It is necessary to calculate its apothem if it is known that the area of this triangle is 34 cm2, 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=√3/4a2
Where we get the length of side a:
a=2√(S/√3)
To determine the apothem hbwe use the formula containing the side edge b. In the case under consideration, its length is equal to the length of the base, we have:
hb=√(b2- a2/4)=√3/2 a
Substituting the value of a through S,we get the final formula:
hb=√3/22√(S/√3)=√(S√3)
We got a simple formula in which the apothem of a pyramid depends only on the area of its base. If we substitute the value S from the condition of the problem, we get the answer: hb≈ 7, 674 cm.
