When studying the characteristics of spatial figures in the course of stereometry, much attention is paid to such properties as area and volume. At the same time, it is important to know the linear parameters of the figures in order to be able to calculate the indicated properties. In this article, we answer the question of how to find the apothem of the pyramid of the regular triangular.
Which figure will be considered?
A triangular pyramid with a regular base is a figure in space, which is limited by one equilateral triangle (base) and three isosceles triangles (sides). To be able to more clearly represent this pyramid, we will show it in the figure.
An important point of any pyramid is its top, which does not belong to the base. If you lower the perpendicular from it to the base, then its length will be the height of the figure. In what follows, we denote the height by the letter h. The height of the regular pyramid falls exactly at the geometric center of the triangle (the intersection point of its medians, as well as bisectors and heights). The second linear parameter that you should know is the length of the side of the base of the triangular pyramid, that is, the length of the side of an equilateral triangle. Denote it by the letter a.
The triangular pyramid has its own name - a tetrahedron. The tetrahedron is not a purely theoretical geometric figure. It is also found in some natural structures. So, in a diamond, a carbon atom is connected to four of the same atoms that make up the tetrahedron. Another example is a methane molecule, in which carbon, connected to four hydrogen atoms, forms a regular triangular pyramid.
The formula of the apothem of the pyramid of the regular triangular
We proceed directly to the question of the article. For a triangular pyramid, the correct apotheme is any of the heights of the side triangles, omitted from the top of the figure. Denote it by h b . Since the figure in question consists of three side triangles that are equal to each other, it has three identical apophemes h b .
Determining the length of the apothem is not difficult. Suppose that the height h and the length of the side a are known. Draw the height of the figure and consider the right triangle, which is located inside the pyramid and is formed by the following sides:
- apophema h b (hypotenuse);
- height h (one leg);
- 1/3 of the median m of an equilateral triangle (second leg).
The length of the median m of the triangle at the base is:
m = √3 / 2 * a
Using the Pythagorean theorem, we obtain the formula for the apotheme length h b :
h b = √ ((1/3 * m) 2 + h 2 ) =>
h b = √ (a 2/12 + h 2 )
This formula shows that the length of the apotheme h b for any parameters of the triangular pyramid is always greater than its height h.
The solution to the problem of determining the value of hb
We will solve an interesting problem. We calculate the length of the apothem for the tetrahedron, in which all the edges are equal to each other.
Denote the length of the edge by the letter a. She is the side of the triangle at the base. To determine h b , you need to find h. It is not difficult to do this if we consider a right-angled triangle formed by a height h, an edge a, and two-thirds of the median m. We get:
h = √ (a 2 - 4/9 * m 2 ) = √ (a 2 - 4/9 * 3/4 * a 2 ) = a * √ (2/3)
Now we apply the formula for apothem, we get:
h b = √ (a 2/12 + h 2 ) = √ (a 2/12 + 2/3 * a 2 ) = √3 / 2 * a
We got the obvious result. The apothem of the regular triangular pyramid is equal to the length of the median of any of the equilateral triangles.