Volumes of Frustrum of Cone, Rectangular-Based Pyramid and Other Pyramid

Welcome to class!

In today’s class, we will be talking about Volumes of a frustum of cone, rectangular-based pyramid and other pyramids. Enjoy the class!

Volumes of Frustum of Cone, Rectangular-Based Pyramid and Other Pyramid

Volumes of Frustum Of Cone

A frustum may be formed from a right circular cone by cutting off the tip of the cone with a cut perpendicular to the height, forming a lower base and an upper base that are circular and parallel. The problem can be generalized to other cones and n-sided pyramids but for the moment consider the right circular cone.

Let h be the height, R the radius of the lower base, and r the radius of the upper base. One picture of the frustum is the following.

Given R, r, and h, find the volume of the frustum.

Conical Frustum Shape (of a right circular cone)

r1 = radius1

(R)r2 = radius2

h = height

s = slant height

V = volume

L = lateral surface area

T = top surface area

B = base surface area

A = total surface area

π = pi = 3.1415926535898

√ = square root

In the combination of solids, we added the volumes of two adjoining shapes which gave us the total volume of any structure. But for frustum of the cone as we are slicing the smaller end of the cone as shown in the figure, hence we need to subtract the volume of the sliced part.

The volume of the frustum of a cone

The frustum as said earlier is the sliced part of a cone, therefore for calculating the volume, we find the difference of volumes of two right circular cones.

From the figure, we have, the total height H’ = H+h and the total slant height L =l1 +l2. The radius of the cone = R and the radius of the sliced cone = r.

Now on seeing the whole cone with the sliced cone, we come to know that the right angle of the whole cone Δ QPS is similar to the sliced cone Δ QAB.

Rectangular-Based Pyramid

A rectangular pyramid is a three-dimensional object with a rectangle for a base and a triangular face corresponding to each side of the base. The triangular faces which are not the rectangular base are called lateral faces and meet at a point called the vertex or apex. Usually, right pyramids are studied.