The main characteristic of any geometric figure in space is its volume. In this article, we will consider what a pyramid with a triangle at the base is, and also show how to find the volume of a triangular pyramid - the correct full and truncated one.
What is a triangular pyramid?
Everyone has heard of the ancient Egyptian pyramids, however, they are regular quadrangular, not triangular. We will explain how to get a triangular pyramid.
Take an arbitrary triangle and connect all its vertices with some one point located outside the plane of this triangle. An educated figure will be called a triangular pyramid. It is shown in the figure below.
As you can see, the figure in question is formed by four triangles, which are generally different. Each triangle is the sides of the pyramid or its face. This pyramid is often called a tetrahedron, that is, a tetrahedral volumetric figure.
In addition to the sides, the pyramid also has edges (there are 6 of them) and vertices (there are 4 of them).
Regular pyramid with a triangular base
A figure that is obtained using an arbitrary triangle and a point in space will be an irregular inclined pyramid in the general case. Now imagine that the original triangle has the same sides, and the point of space is located exactly above its geometric center at a distance h from the plane of the triangle. A pyramid constructed using this source data will be correct.
Obviously, the number of edges, sides, and vertices of a regular triangular pyramid will be the same as that of a pyramid constructed from an arbitrary triangle.
However, the correct figure has some distinguishing features:
- its height drawn from the top will exactly cross the base at the geometric center (the intersection point of the medians);
- the side surface of such a pyramid is formed by three identical triangles, which are isosceles or equilateral.
A regular triangular pyramid is not only a purely theoretical geometric object. Some structures in nature have its shape, for example, a diamond crystal lattice, where a carbon atom is connected to four of the same atoms by covalent bonds, or a methane molecule, where the vertices of the pyramid are formed by hydrogen atoms.
Volume formulas of a triangular pyramid
You can determine the volume of absolutely any pyramid with an arbitrary n-gon in the base using the following expression:
V = 1/3 × S o × h
Here, the symbol S o denotes the area of the base, h is the height of the figure drawn to the marked base from the top of the pyramid.
Since the area of an arbitrary triangle is equal to half the product of the length of its side a by the apothem h a dropped to this side, the formula for the volume of a triangular pyramid can be written in the following form:
V = 1/6 × a × h a × h
For a triangular pyramid of a general type, determining the height is not an easy task. To solve it, the easiest way is to use the distance formula between a point (vertex) and a plane (triangular base), represented by an equation of a general form.
For the correct pyramid, the volume formula has a specific form. The area of the base (equilateral triangle) for it is equal to:
S o = √3 / 4 × a 2
We substitute it into the general expression for V, we obtain:
V = √3 / 12 × a 2 × h
A special case is the situation when the tetrahedron has all sides turned out to be equal equilateral triangles. In this case, its volume can be determined only on the basis of knowledge of its edge parameter a. The corresponding expression is:
V = √2 / 12 × a 3
Truncated pyramid
If the upper part containing the vertex is cut off at the regular triangular pyramid, a truncated figure will be obtained. Unlike the original, it will consist of two equilateral triangular bases and three isosceles trapezoids.
The photo below shows how a regular truncated triangular pyramid made of paper looks like.
To determine the volume of a truncated triangular pyramid, it is necessary to know its three linear characteristics: each of the sides of the bases and the height of the figure, equal to the distance between the upper and lower bases. The corresponding formula for the volume is written as follows:
V = √3 / 12 × h × (A 2 + a 2 + A × a)
Here h is the height of the figure, A and a are the lengths of the sides of the large (lower) and small (upper) equilateral triangles, respectively.
The solution of the problem
To make the information presented in the article more understandable for the reader, we will show by a clear example how to use some of the written formulas.
Let the volume of the triangular pyramid be equal to 15 cm 3 . It is known that the figure is correct. You should find the apothem a b of the side rib, if it is known that the height of the pyramid is 4 cm.
Since the volume and height of the figure are known, one can use the corresponding formula to calculate the length of the side of its base. We have:
V = √3 / 12 × a 2 × h =>
a = 12 × V / (√3 × h) = 12 × 15 / (√3 × 4) = 25.98 cm
Apothem a b can be calculated by considering the corresponding right-angled triangle inside the pyramid. The legs of the triangle are 1/3 of the length of the height of the base and the height of the pyramid, the desired apothem will be the hypotenuse. Then:
a b = √ (h 2 + a 2/12) = √ (16 + 25.98 2/12) = 8.5 cm
The calculated length of the apothem of the figure turned out to be greater than its height, which is true for a pyramid of any type.