The three-dimensional figure that often appears in geometric problems is a pyramid. The simplest of all figures in this class is triangular. In this article, we analyze in detail the basic formulas and properties of a regular triangular pyramid.
Geometric representation of the figure.
Before proceeding to consider the properties of a regular triangular pyramid, we will examine in more detail what kind of figure we are talking about.
Suppose that there is an arbitrary triangle in three-dimensional space. In this space, we select any point that does not lie in the plane of the triangle, and connect it with the three vertices of the triangle. We got a triangular pyramid.
It consists of 4 sides, all of which are triangles. Points at which three faces join are called vertices. The figures also have four. The lines of intersection of two faces are edges. The edges of the pyramid under consideration 6. The figure below shows an example of this figure.
Since the figure is formed by four sides, it is also called the tetrahedron.
Regular pyramid
Above, an arbitrary figure with a triangular base was considered. Now suppose that we have drawn a perpendicular segment from the top of the pyramid to its base. This segment is called height. Obviously, you can draw 4 different heights for the figure. If the triangular base intersects the geometric center in the geometric center, then such a pyramid is called a straight line.
A straight pyramid, the base of which will be an equilateral triangle, is called regular. For her, all three triangles forming the lateral surface of the figure are isosceles and are equal to each other. A special case of a regular pyramid is a situation where all four sides are equal equilateral triangles.
We consider the properties of a regular triangular pyramid and give the corresponding formulas for calculating its parameters.
Base side, height, side rib and apothema
Any two of the listed parameters uniquely determine the other two characteristics. We give the formulas that bind these quantities.
Assume that the side of the base of the triangular pyramid is equal to a. The length of its side rib is b. What will be the height of the regular triangular pyramid and its apothema.
For height h we get the expression:
h = √ (b 2 - a 2/3)
This formula follows from the Pythagorean theorem for a right-angled triangle whose sides are the side edge, height and 2/3 of the height of the base.
Apotema pyramid is called the height for any lateral triangle. The length of the apotheme a b is equal to:
a b = √ (b 2 - a 2/4)
It can be seen from these formulas that, whatever the side of the base of the pyramid is triangular regular and the length of its side edge, the apothema will always be greater than the height of the pyramid.
The presented two formulas contain all four linear characteristics of the figure in question. Therefore, using the known two of them, one can find the rest by solving a system of written equalities.
Figure volume
For absolutely any pyramid (including an inclined one), the value of the volume of space limited by it can be determined by knowing the height of the figure and the area of its base. The corresponding formula is:
V = 1/3 * S o * h
Applying this expression for the figure in question, we obtain the following formula:
V 3 = √3 / 12 * a 2 * h
Where the height of the regular triangular pyramid is h and its base side is a.
It is not difficult to obtain a formula for the volume of a tetrahedron, in which all sides are equal and represent equilateral triangles. In this case, the volume of the figure is determined by the formula:
V = √2 / 12 * a 3
That is, it is determined uniquely by the length of the side a.
Surface area
We continue to consider the properties of the triangular regular pyramid . The total area of all faces of a figure is called its surface area. The latter is convenient to study, considering the corresponding scan. The figure below shows what a scan of a regular triangular pyramid looks like.
Suppose that we know the height h and the side of the base a of the figure. Then the area of its base will be equal to:
S o = √3 / 4 * a 2
Each student can get this expression if he remembers how to find the area of a triangle, and also takes into account that the height of an equilateral triangle is also a bisector and a median.
The area of the lateral surface formed by three identical isosceles triangles is:
S b = 3/2 * √ (a 2/12 + h 2 ) * a
This equality follows from the expression of the apotheme of the pyramid through the height and length of the base.
The total surface area of the figure is:
S = S o + S b = √3 / 4 * a 2 + 3/2 * √ (a 2/12 + h 2 ) * a
Note that for a tetrahedron in which all four sides are the same equilateral triangles, the area S will be equal to:
S = √3 * a 2
Properties of a regular truncated triangular pyramid
If the considered triangular pyramid has a plane cut parallel to the base, then the remaining lower part will be called a truncated pyramid.
In the case of a regular pyramid with a triangular base, as a result of the described section method, a new triangle is obtained, which is also equilateral, but has a shorter side length than the side of the base. A truncated triangular pyramid is shown below.
We see that this figure is already bounded by two triangular bases and three isosceles trapezoids.
Suppose that the height of the resulting figure is equal to h, the lengths of the sides of the lower and upper bases are a 1 and a 2, respectively, and the apothema (trapezoid height) is a b . Then the surface area of the truncated pyramid can be calculated by the formula:
S = 3/2 * (a 1 + a 2 ) * a b + √3 / 4 * (a 1 2 + a 2 2 )
Here, the first term is the area of the lateral surface, the second term is the area of triangular bases.
The volume of the figure is calculated as follows:
V = √3 / 12 * h * (a 1 2 + a 2 2 + a 1 * a 2 )
To unambiguously determine the characteristics of a truncated pyramid, it is necessary to know its three parameters, as shown by the above formulas.