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Conical Frustum
A Conical Frustum is a Frustum created by slicing the top off a cone (with the cut made parallel to the base), forming a lower base and an upper base that are circular and parallel. Let be the height, the radius of the lower base, and the radius of the upper base as pictured below:
The Volume of Frustum:
The Volume of the Frustum could be found using the formula:
Now, let’s derive the formula without using calculus.
Consider the cone before it was cut. Let the height of the cut be.
The volume of the cone,
The volume of the original pre-cut cone ,
The volume of the cut part,
Now, to get the volume of the frustum ( ), we have to subtract the volume of the cut part from the volume of the original cone. So,
Now, consider the original pre-cut cone. The triangles formed by the height and the bases are similar by similarity and the sides are proportional.
Hence,
So we have,
Substitute in equation , we have
Hence the formula of volume of the Frustum.
Total Surface Area of a Frustum:
Now, let’s examine how to find the Surface Area of the frustum.
We know the lateral area of a right circular cone is. For the right circular cone here, let be the slant height and be the top and bottom radii. Then,
And, from the figures:
The length of arc is the circumfence of the base:
Again, from the figure:
Hence,
Volume of a truncated Pyramid with a square base:
Let, the area of the lower base
the area of the upper base
perpendicular distance between and (also known as the atitude of frustum)
Note here that and are parallel to each other.
Now, Volume of frustum,
And top of cone,
So,
