The relationship between the angle of a base-anchored cross section and the volumes of truncated n-sided regular prisms and spheres
Abstract – This paper will aim to investigate different shapes that are truncated in 3-D coordinate geometry such as regular triangular prisms, cylinders, and spheres which links to other topics in mathematics such as trigonometry, functions, and calculus. My research question is: “to what extent can the volume of spheres and n-sided regular prisms be described based on the angle of their cross section with the base?” The base is the area highlighted in red in the diagram on the right, the cross section is the plane highlighted in blue, the angle of their cross section is the angle highlighted in green and the truncated volume is therefore the volume under the cross section. First, I will solve the original competition question (from the 2019 Chinese National High School Math League Paper attached in the appendix) Then I will extend my original question to investigate the relationship between the angle of the base-anchored cross section and the volume of the truncated cube. Following the extension, I will investigate the relationship between the angle of the base-anchored cross section to the volume of n-sided regular prism and for a sphere. This will be explored in two different parts as the conditions change when the angle of the base-anchored cross section meets the volume above and below the height. Then I will use actual values to substitute inside my equations to find the ratio of the volume under a given base-anchored angle to the total volume. Finally, I will compare the answers for different regular prisms and attempt to find a general equation describing the relationship between the angle of a base-anchored cross section and the truncated volume for n-sided regular prisms.