# Thread: Find the volume of the described solid.

1. ## Find the volume of the described solid.

A pyramid with height h and base an equilateral triangle with side a (a tetrahedron).

This is an example problem from the text. If someone could walk me through how to do this problem using integration, I think it will help me grasp how to do the other problems.

I'm having trouble starting, and not sure what I'm supposed to do. I think I'm supposed to find the surface area of the triangle with respect to height, and evaluate it from 0 to h. But how to I find out what the triangular base is at any given height?

2. ## Cross-Sectional Area

Imagine the center of the equilateral triangle base sitting on the origin in the xy-plane. The figure comes to a point somewhere along the z-axis at the point (0,0,h).

The general formula for finding volume of such a figure is $V=\int_a^bA(r)dr$, where $A(r)$ is the cross-sectional area of the figure at a distance r from the origin. Here, your bounds of integration will be $0\to h$. $A(h)=0$ since it comes to a point, and $A(0)=\frac{\sqrt3}2a^2$ since this is the area of an equilateral triangle with side length $a$.

Since the side-length of the equilateral cross-sectional triangle shrinks linearly from a to 0, it can be represented as $s(r)=\frac{h-r}{h}a$, so the area $A(r)=\frac{\sqrt3}2a^2\frac{(h-r)^2}{h^2}$

And therein lies your integral: $V=\frac{\sqrt3}2\frac{a^2}{h^2}\int_0^h(h-r)^2dr$

(You can of course check your work by using the simple geometric formula $V=\frac13bh$ for figures that start at a base of area b and rise to a point at height h.)