TRUNCATED PYRAMID

Аннотация

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 ofatruncated pyramid in various fields. Keywords: Truncated pyramid, geometry, triangles, surface area calculation, volumecalculation, geometricbodies. 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 methodsinclude, for example, estimating thevolumeby weighing models of atruncated 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 asimilar formulain theclassic work ofChinese mathematics, theNine Chapters. These approaches are not only important for historical-mathematical analysis, but also show the level of ancient mathematical thinking and theempirical-practical approaches inherent in them in the process of understanding the volume of the truncated pyramid and finding a reliable formula. 96 Volume 4, issue 5, 2025 This serves as an important source for in-depth study of the topic of the truncated pyramid in a modern andhistorical context.1 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 theamount of materials needed for construction, as well as thetime spent onthe 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 isapproximately one-third thevolumeofthesquare prism witha basethat forms it. Expressed in mathematical language, this means the following formula: if the base of a pyramid has an areaof ​ ​ a2and aheight of h,then itsvolumeis: V =1/3 *a2*h P 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 volumeofthe 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 * a2* h - serves as a reference point for entering the formula forthetruncated 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 onlyin theoretical mathematics, but also in engineering, architecture, andother 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 th