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