Volume of a Truncated Cone via Geometric Similarity

Blagovest Ivanov
164th High School „Miguel de Cervantes“, Sofa, Bulgaria

https://doi.org/10.53656/math2026-1-3-vtc

Abstract. The following study addresses the issue of calculating with precision the volume of a truncated right circular cone while provided with limited information on the dimensions of the object itself (being given the relation between the radii, the vertical heights, or the slanted heights). The results include the proof of two theorem generalizations for the calculation of said volume with either of the three given elements via the principles of geometric similarity. It is shown that, due to the similarity between the full cone and the smaller removed cone, the volume of the truncated cone can be expressed using the dierence of cubes of the corresponding linear dimensions. Revisiting the classical volume formula through the principles of geometric similarity, this work provides six alternative expressions that have both theoretical value and direct applications, especially in the field of education.

Keywords: Right circular truncated cone, Volume formulas, Geometric similarity, Multi-dimensional parameterization, Educational, applied, and theoretical value

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