A Note on Isoperimetric Problems in Solid Figures Andri Setiawan (a), Denny I. Hakim (b*), Oki Neswan (b)
(a) Student at Magister Pengajaran Matematika, Institut Teknologi Bandung, Indonesia
(b) Lecturer at Institut Teknologi Bandung, Indonesia
*Corresponding Author
Abstract
In this paper, we explore some extension of isoperimetric problems in solid figures, focusing on oblique and right prisms with rectangular, triangular, and regular hexagonal bases. Isoperimetric problems in solid figures involve finding the shape or configuration that maximizes or minimizes a measure (volume of a solid) subject to a constraint on another measure (surface area).
In the case described, the problem is to find the prism with the maximum volume given a fixed surface area. Through simple algebraic and trigonometric manipulations, we obtain proof that a right prism will provide a larger volume than an oblique prism with the same surface area. Since the area of the base is the same for both prisms, we can compare their heights to determine which one has the larger volume. For a right prism, the height is simply the distance between the base and the top of the prism. For an oblique prism, the height is the distance between the base and the top of the prism projected onto the base. This projected height will always be shorter than the actual height of the prism, so the volume of an oblique prism will always be smaller than that of a right prism with the same base area and surface area.
Our result can be extended to other polygonal bases as well, not just rectangles, triangles, and hexagons. It can also be generalized to other types of solids, such as cylinders and cones. Additionally, it can be utilized as a material for the enrichment of geometry topics in math education.
Keywords: Isoperimetric, Solid Figures, Isoperimetric Problems in Solid Figures
