The volume of a pyramid with height h and base an equilateral triangle with side a.
Definition of volume:
Let S be a solid that lies between x=a and x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is
Height of pyramid is h and basean equilateral triangle with side a.
Consider the triangle consisting of two vertices of the base and the center of the base.
This triangle is similar to the corresponding triangle at a height y,
So from the figure,
Also, by similar triangles,
These two equations imply that
Since the cross section is an equilateral triangle, it has area
After substituting the value of Α,
So the volume is,
V= ∫0hAydy= ∫0h34a21-yh2dy
After expanding the square term,
Applying the Fundamental Theorem of Calculus,
The volume of a pyramid with height h and base an equilateral triangle with side a is 312a2h