Volume with Cross Sections
Proving Formula for Pyramid

This video show you how to use calculus to prove the volume formula for a pyramid with a square base using cross-sections and integration.

Summary

In this video, you'll see how to use Calculus to prove the volume formula for a Pyramid. By slicing the pyramid perpendicular to its height, each cross-section forms a square whose side length changes with x. Writing the area of each slice as a function of x and integrating from the vertex to the base shows why the pyramid volume formula is V = (1/3)Bh, where B is the area of the base and h is the height. This approach reinforces connections between geometry and integral calculus.

Key Takeaways

• A pyramid’s volume can be derived using calculus and cross-sections.

• The pyramid is positioned with its vertex at the origin and its height along the positive x-axis.

• Each cross-section perpendicular to the x-axis is a square.

• Similar triangles relate the side length of a square slice to its distance x from the vertex.

• The side length of a slice is (base side length / height) times x.

• The area of a slice is proportional to x squared.

Transcript

Introduction

Hey guys, Mr. Antonucci here, and we’re going to use calculus to prove the volume formula for a pyramid with a square base. 

Up Next: Volume with Cross Sections: Proving Formula for a Cone