Expert: Socrates - 10/1/2009

Question
Hello,

The question is: Find the volume of a pyramid with height (h) and with a base of an equilateral triangle with side (a).

I am unclear on how to apply this to the integration system.  I know how to do volume and area using equations but this just blows my mind. Any help would be greatly appreciated.


Thanks,
Andrew

Answer
Use integration to "sum up" cross section areas to find the volume.
A horizontal plane cross section of this pyramid will be an equilateral triangle.

Use similar triangles and draw a diagram if necessary to convince yourself that at height x , the plane cross section will be an equilateral triangle with side length (a)(h-x)/h

The area of an equilateral triangle is s^2 √3 /4 , where s is the length of a side.
So our horizontal cross section has area √3 a^2 (h-x)^2 / 4h^2

Sum up these areas from x=0 to x=h by integrating √3 a^2 (h-x)^2 / 4h^2  from 0 to h with respect to x

Factor out the constants ,  √3 a^2 / 4h^2  S  (h-x)^2 dx

an anti derivative for (h-x)^2 is -1/3 (h-x)^3

Evaluate at x=h , x=0 , then find the difference to be (1/3)h^3

Multiply by the constants that were factored out of the integeral and get

(1/3)h^3 √3 a^2 / 4h^2 = ha^2/4√3


The volume is ha^2/4√3