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TRUNCATED PYRAMID
Mamaraimov Bekzod Kadirovich
Teacher of mathematics at Terdu Academic Lyceum.
Makhmudov Azam Kudratovich
Terdu Academic Lyceum Mathematics teacher.
Musurmonov Maruf Akrom ugli
mathematics teacher at Terdu Academic Lyceum.
Abstract:
This article analyzes the concept of a truncated pyramid, its main properties and its
place in geometry. The geometric properties of a truncated pyramid, formulas for calculating its
surface area and volume are considered in detail. Also, brief information is provided about the
application of a truncated pyramid in various fields.
Keywords:
Truncated pyramid, geometry, triangles, surface area calculation, volume calculation,
geometric bodies.
The main goal of teaching geometry is to systematically teach students the basic properties of
plane and spatial shapes. By forming the skills to solve mathematical and constructive problems
that are solved through the calculation methods of these properties, it is intended to expand the
spatial imagination of students and develop their logical thinking. In addition, it is important to
form the skills to apply theoretical knowledge acquired in geometry lessons in real-life tasks
such as measuring the surface of the earth, determining the areas and volumes of various
technical devices, and performing geometric calculations necessary in engineering and
construction. Thus, geometry is one of the important disciplines that provides not only
theoretical knowledge, but also practical skills, deepening the student's thinking. Some historians
have put forward various empirical, intuitive, and relatively more formal methods for how the
Egyptians could have found formulas for calculating the volume of a truncated pyramid. These
methods include, for example, estimating the volume by weighing models of a truncated pyramid,
studying it as a cube, four prisms, and small pyramids at the four corners, understanding the
formula as based on the average of three volumes or three surfaces, and extracting a simple
arithmetic operation such as h/3 as a common factor. However, none of these works explained
the formula by rearranging three identical copies of the truncated pyramid (i.e., by eliminating
the pyramids at the 12 corners and using their volume—as material—to construct the four-
cornered cuboids in the largest of the three resulting boxes). Interestingly, in the 1970s, scholars
studying Chinese mathematics discovered that this method was largely similar to a method used
by the Chinese mathematician Liu Hui in the 3rd century AD. Liu Hui used this method to prove
a similar formula in the classic work of Chinese mathematics, the Nine Chapters.
These approaches are not only important for historical-mathematical analysis, but also show the
level of ancient mathematical thinking and the empirical-practical approaches inherent in them in
the process of understanding the volume of the truncated pyramid and finding a reliable formula.
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This serves as an important source for in-depth study of the topic of the truncated pyramid in a
modern and historical context.
Also, according to Egyptologists, the construction of pyramids, especially symmetrical and
perfectly built pyramids, was one of the most important engineering tasks for the ancient
Egyptians. This activity lasted for centuries, required huge financial costs, involved wide
segments of the population, and required extremely precise planning and advance calculations of
the amount of materials needed for construction, as well as the time spent on the work. Therefore,
the people responsible for these works had certain knowledge about the volume of the pyramid,
that is, facts based on practical experience. In particular, they knew that the volume of a pyramid
is approximately one-third the volume of the square prism with a base that forms it.
Expressed in mathematical language, this means the following formula: if the base of a pyramid
has an area of a
2
and a height of h, then its volume is:
V
P
= 1/3 * a
2
* h
This formula is called the “indirect formula for the volume of the Egyptian pyramid” in modern
terminology. This formula shows that although the ancient Egyptians did not write down precise
mathematical formulas, they deeply understood the essence of such equations through their
practical experience and architectural developments. This knowledge later served as the basis for
calculating the volume of the truncated pyramid.
A truncated pyramid is a form of a regular pyramid with the upper part cut off, and the same
approach was used to calculate its volume, that is, to determine the difference between two
complete pyramids or to calculate it using special formulas. Therefore, the formula for the
volume of a regular pyramid –1/3 * a
2
* h - serves as a reference point for entering the formula
for the truncated pyramid.
In geometry, the pyramid shape is one of the most studied objects. The truncated shape of a
pyramid, that is, a pyramid with its upper part cut off, is called a truncated pyramid. It is widely
used not only in theoretical mathematics, but also in engineering, architecture, and other practical
fields.
A pyramid is a three-dimensional geometric object. The base of a pyramid is formed by a flat
polygon. A point that does not lie in the plane of the base, that is, is located above the base, is
called the apex of the pyramid. The intersections between the apex of the pyramid and all the
vertices of its base form the edges of the pyramid.
Since the base of a pyramid can be any polygon, the shape itself depends on this. If its base is a
triangle, it is called a triangle. Depending on the base, pyramids can be rectangular, pentagonal,
hexagonal, etc.
If the problem does not know which polygon lies at the base of the pyramid, or if it is not
important, then the condition calls such a pyramid n-gonal.
1
Putilov I.I., Sidorov I.B. Стереометрия: Учебник для 10–11 классов. – Москва: Просвещение, 2019.
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The lines extending from the apex of the pyramid to the apex of the base are called lateral edges.
The surface of the pyramid is formed by the main surface formed by the base plane and the
lateral surfaces formed by the lateral edges. Each lateral surface is a triangle, one side of which is
the apex of the pyramid, and the opposite side is a side of the base.
To measure the distance between the apex of the pyramid and the base plane, a perpendicular is
drawn from three to the base. This perpendicular is called the height of the pyramid.
If the number of sides of the polygon forming the base of the pyramid is n, then such a pyramid
is called an n-gonal pyramid. For example, if the base is a triangle, then such a pyramid is called
a triangular pyramid or, by another name, a tetrahedron.
A truncated pyramid is a part formed by cutting off the top of two parallel-base pyramids of the
same shape. For example, if we cut off the top of the pyramid, the remaining part will be a
truncated pyramid.
When a pyramid is cut with a plane parallel to its base, two polyhedra are formed. One of them is
called a truncated pyramid, and the other is a pyramid, which is called the complement of the
truncated pyramid. The bases of a truncated pyramid consist of similar polygons, and the sides
are trapezoids. The height of a truncated pyramid is called the perpendicular section whose ends
are at the bases. If a truncated pyramid is part of a regular pyramid, it is called a regular
truncated pyramid. The sides of a regular truncated pyramid consist of equal-sided trapezoids.
The height of these trapezoids is called the apotheme of the regular truncated pyramid. The
lateral surface area of a regular truncated pyramid is calculated using the following formula:
S
yon
= (P
1
+P
2
) * h
The volume of a regular truncated pyramid is calculated using the following formula:
V= ( S
1
+S
2
) * H
The total surface area of a truncated pyramid consists of the following parts:
- Surface area of the large base
- Surface area of the small base
- Area of the lateral surfaces
The formula for calculating the total surface area is:
S = S₁ + S₂ + S
yon
A truncated pyramid is one of the spatial shapes that plays an important role in geometry. It is a
shape formed when the top of a regular pyramid is cut by a horizontal plane, and consists of two
bases (upper and lower) and lateral edges. The study and practical application of such a shape is
important in several areas.
The truncated pyramid serves as a means of teaching important concepts among spatial figures.
Through this shape, students:
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Deeply master basic geometric concepts such as volume and surface area of their div;
Create an idea of the similarity, proportionality and ratios between objects in space;
Form the skills of geometric modeling, analysis and the use of formulas.
In addition, the fact that the truncated pyramid appears as a shape of many real objects in life
indicates its practical importance. In particular:
In architecture and construction: Elements in the shape of a truncated pyramid are widely used in
towers, columns, foundations, steps and roof structures. This ensures their stability, aesthetic
appearance and functionality.
In engineering and industry: Machine parts, ventilation holes, containers, light bulb covers and
many other items are designed in the shape of a truncated pyramid.
In the field of art and design: This shape gives a beautiful and harmonious look to objects such as
architectural elements, sculptures, decorations, lamp stands.
The truncated pyramid shape has a symmetrical, harmonious and balanced appearance and is
widely used by designers to create decorations. It seems beautiful and natural to the human eye,
therefore it occupies a special place in interior and exterior design.
The truncated pyramid also provides students with important skills in solving practical problems:
• Calculating volume:
• Finding surface areas;
• Used in mathematical modeling, analysis and graphic design.
As a tool that helps students develop spatial imagination, mathematical thinking and simplify the
study of complex objects, the truncated pyramid occupies a special place in the science of
geometry. It serves to improve the quality of education by combining educational, advanced and
practical knowledge.
It is also worth noting that Geometry is one of the important subjects that form spatial
imagination, logical thinking and problem analysis skills in students. Today, the effectiveness of
the learning process can be increased by using modern approaches, innovative technologies and
interactive methods in teaching geometry.
Through the STEAM approach, this is a method of teaching science (S), technology (T),
engineering (E), art (A) and mathematics (M) in an interconnected way. This method allows
students to connect theory with real-life practice.
Application for the truncated pyramid: students analyze structures in the field of architecture. For
example, they study this topic through truncated pyramid shapes in the form of historical
pyramids, water tanks or sculptures.
Also, the use of digital tools, including GeoGebra, AutoCAD, or other 3D modeling programs,
allows students to better understand geometric shapes in space.
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For the truncated pyramid: the program explains the topic in more depth by creating a complete
pyramid and its top cutaway view, and automatically calculating their volume and surface area.
In addition, through these methods, students make geometric shapes using paper, plastic or other
materials. This develops their manual labor and spatial imagination.
For the truncated pyramid: students make a truncated pyramid from cardboard or other material.
Then, through it, they practically measure the lateral surface, base surface, height and volume.
Then, through the Problem-Based Learning method, students independently search, think and try
to find a solution.
For the truncated pyramid: for example, by solving the problem “A container for pouring water
is in the shape of a truncated pyramid. How can its full volume be determined?”, the topic is
reinforced.
In conclusion, the truncated pyramid is one of the important objects studied in geometry. The
formulas for calculating its surface area and volume are widely used not only in theoretical but
also in solving practical problems. An in-depth study of the properties and structure of the
truncated pyramid is scientifically and practically useful.
References:
1.
Karimov A., Tursunov R. Geometriya: umumiy o‘rta ta’lim maktablari uchun darslik (9-
sinf). – Toshkent: O‘qituvchi, 2020.
2.
Putilov I.I., Sidorov I.B. Стереометрия: Учебник для 10–11 классов. – Москва:
Просвещение, 2019.
3.
Thomas, G.B., Finney, R.L. Calculus and Analytic Geometry. – Addison Wesley, 2002.
(Bo‘lim: Solids and Volumes – kesik piramidalar haqida)
4.
Stewart, J. Calculus: Early Transcendentals. – Cengage Learning, 8th edition, 2015.
