To determine
To find:
The volume of a pyramid with height h and base an equilateral triangle with side a.
Answer
312a2h.
Explanation
1) Concept:
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
V=limn→∞∑i=1nAxi*∆x=∫abAxdx
2) Given:
Height of pyramid is h and basean equilateral triangle with side a.
3) Calculations:

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,
ab=ΑΒ →Α=aΒb.
Also, by similar triangles,
bh=Βh-y→ Β=bh-yh.
These two equations imply that
Α=a(1-yh)
Since the cross section is an equilateral triangle, it has area
Ay=12·Α·32Α
After substituting the value of Α,
Ay=34·a21-yh2
So the volume is,
V= ∫0hAydy= ∫0h34a21-yh2dy
After expanding the square term,
V=34a2∫0h1-2yh+y2h2dy
After integrating,
V=34a2y-y2h+y33h2h0
Applying the Fundamental Theorem of Calculus,
V=34a2h-h+h3-0
After simplifying,
V=34a2·h3=312a2h.
Conclusion:
The volume of a pyramid with height h and base an equilateral triangle with side a is 312a2h