One of the symmetric polyhedra whose properties are studied by stereometry is a pyramid. In this article, we consider in more detail the following questions: what is a truncated pyramid, how can it be obtained, and what properties is it characterized.
Full pyramid
Before revealing the question of what is a truncated pyramid, a general definition of a pyramid should be given.
Under a pyramid in geometry we mean a figure in three-dimensional space, which consists of n triangular faces and one n-angular side, which is called the base. Imagining a pyramid is quite simple: you need to mentally connect all the corners of the n-gon with some one point in space. The figure below shows the figure that is obtained.
Here we see that the corners of the quadrangular base are connected by segments to one point, which is called the top of the pyramid. The lateral surface of the figure is formed by four different triangles.
If all the triangles of the side surface are the same and isosceles, then such a figure is called a straight pyramid. If, moreover, the base will be a regular n-gon, for example, a square, then they speak of a regular pyramid.
Truncated pyramid
The above figure is called a full pyramid. Now we show what a truncated pyramid is and how it can be obtained from the complete one.
Let us have a complete figure with a pentagonal base. It is shown below in the figure to the left.
Take an arbitrary plane and cut off the upper part of the full pyramid. The plane will divide the original figure into two parts: the upper one will also represent the pyramid, but the lower one is the truncated pyramid (see the right image in the figure).
Note that in this case we have selected a secant plane that is parallel to the base of the original shape. A truncated pyramid obtained from a regular shape using a parallel section will also be called regular.
The figure also shows that the bases of the truncated pyramid (pentagons in the example) are formed by similar regular polygons, while the size of the top will always be smaller than the bottom. The lateral surface of this figure, in contrast to the full pyramid, is formed by isosceles trapezoids.
If an n-gon lies at the base of the truncated pyramid, then it has 2 × n vertices, 3 × n edges and n + 2 sides.
Two important geometric parameters of the figure in question are its surface area and volume.
Truncated pyramid surface
Having considered what a truncated pyramid is, we turn to the study of its surface. The latter is understood as the totality of all the faces that make up the figure. The easiest way to study the surface properties is the example of a scan. The figure below shows a scan for a pyramid with pentagonal bases.
To calculate the area of its entire surface, it is necessary to add the area of two bases and the area of all trapezoids. The corresponding formula is:
S = S o1 + S o2 + 1/2 × (P o1 + P o2 ) × A p .
In this expression, the first two terms, i.e., S o1 and S o2 , are the base areas. The third term is the total area of all trapezoids, which is equal to half the product of the sum of the perimeters of the bases P o1 and P o2 by the apothem (height) of the trapezoid A p .
For example, for the case of a quadrangular regular truncated pyramid, this formula will be rewritten in the form:
S 4 = B 2 + b 2 + 2 × (B + b) × A p .
Where B, b are the lengths of the sides of the large and small square bases, respectively.
The volume of the truncated pyramid
To determine the volume of the figure in question, it is necessary to know its height h, as well as the area of both bases S o1 and S o2 . If these characteristics are known, then to determine the volume of the truncated pyramid, you should use the formula:
V = 1/3 × h × (S o1 + S o2 + √ (S o1 × S o2 )).
For example, for a quadrangular regular figure, the lengths of the sides of the bases of which are equal to B and b, we arrive at the following expression for the volume:
V = 1/3 × h × (B 2 + b 2 + B × b).
Problem solving example
Having examined what a truncated pyramid is, and also having figured out the formulas necessary to describe its characteristics, we will show how to use them in practice.
Suppose there is a hexagonal truncated figure, which is shown below.
It is necessary to calculate its volume if the sides of bases B and b and apothem A p are known.
To begin with, we calculate the area of each of the bases, which corresponds to the area of a regular hexagon. We have:
S o1 = 3 × √3 / 2 × B 2 ;
S o2 = 3 × √3 / 2 × b 2 .
To determine the volume, it is necessary to calculate the height h of the figure through A p . Considering the right triangle shown in the figure and applying the Pythagorean theorem, we get:
h = √ (A p 2 - 3/4 × (Bb) 2 ).
Then the volume of this hexagonal truncated pyramid will be equal to:
V = √3 / 2 × √ (A p 2 - 3/4 × (Bb) 2 ) × (B 2 + b 2 + B × b).
Mayan Pyramids
If the Egyptian pyramids in terms of geometry are regular full quadrangular figures, then similar constructions of the Maya Indians are quadrangular truncated pyramids.
These cultural monuments, preserved to this day, once performed a double role for their inhabitants: on the one hand, they served as the tomb of the leaders, on the other hand, on their upper base was a temple where priests worshiped the gods.