Exploring the Properties of Spherical Triangle-Based Pyramids: An Investigation of Geometrical Relationships, Surface Areas, and Volumes

The project aims to delve into the intriguing world of spherical triangle-based pyramids and analyze their unique properties, geometrical relationships, and the impact of curvature on their surface areas and volumes. Spherical triangle-based pyramids are captivating architectural structures formed by placing a triangle on the surface of a spherical cap, resulting in a curved base and three faces leading to the apex. This research seeks to understand the intricate interplay between geometry, curvature, and dimensions, providing valuable insights for various applications, including architecture, engineering, and beyond. Mathematical Foundations: This section lays the groundwork for the investigation by exploring the fundamentals of spherical geometry and spherical triangles. Key concepts like the Law of Sines and Law of Cosines on a sphere will be discussed, as well as the transformation of a spherical cap into a spherical triangle-based pyramid. Geometrical relationships, angles, and side lengths of spherical triangles will be studied, and the equations for surface areas and volumes will be derived. Surface Areas of Spherical Triangle-Based Pyramids: Here, we delve into the calculation methods for surface areas of both regular triangular pyramids (with no curvature) and spherical triangle-based pyramids. We analyze the impact of curvature on surface area calculations, taking into account the offset triangle's curved base and how it influences the final surface area. A comparative analysis between the two methods will provide insights into the effects of curvature on the overall surface area. Volumes of Spherical Triangle-Based Pyramids: The volume calculations of spherical triangle-based pyramids will be investigated in this section. We derive the equations for volume with and without curvature, considering the base's curvature and the unique geometric properties. The comparative analysis between these volume equations will shed light on how the curvature affects the overall volume of the pyramid. Comparative Analysis: The last section offers a comprehensive comparative analysis of the various aspects studied throughout the research. We discuss the findings on geometrical relationships, surface areas, and volumes with and without curvature, highlighting the implications for architectural design and engineering. The impact of curvature on the stability, strength, and overall aesthetics of spherical triangle-based pyramids will be thoroughly explored. Conclusion: The project concludes with a summary of the key findings and their significance. We reflect on the exploration of spherical triangle-based pyramids, uncovering the intricate relationships between geometry, curvature, and dimensions. The implications of the research for architectural structures, design optimization, and real-world applications will be discussed, highlighting potential areas for future exploration. This project is a maths IB Extended essay that I have to do. so far I have most of the math covered and have 3d models that I can use to explain. But some parts I need insight on by mathematical experts please check the file I attached it has my EE so far. read into it and see if they are any problems.