Volume of Frustum of Cone
Frustum of cone is a sliced part of a cone, where a small cone is removed from the larger cone. Therefore, to calculate the volume of the frustum of cone, one just needs to calculate the difference between the volume of the larger and smaller cone.
Let’s assume,
- Total height of the cone is to be H + h
- Total slant height to be l’ + L
- The radius of a complete cone is r
- The radius of the sliced cone is r’
Since the volume of the cone is given as V = 1/3πr2h
Volume of complete cone V1 = 1/3πr2(H+h)
Volume of smaller cone V2 =1/3πr’2(h)
Now the volume of the frustum of cone (V) can be calculated using the formula,
V = V1 – V2
V = 1/3πr2(H+h) – 1/3πr’2(h)
V= 1/3π[r2(H+h) – r’2(h)]…(1)
Using the property of similarity of the triangles of △OCD and △OAB, one can write,
r / (H + h) = r’ / h
r / r’ = (H + h) / h
H + h = hr / r’
Substitute this value of (H+h) in equation (1), and simplify,
V = 1/3π[r2(rh / r’) – r’2(h)}
= 1/3π[{hr3 – hr’3} / r’]…(2)
Using the similar triangle’s property again in △OCD and △OAB, we will find out the value of h
r / (H + h) = r’ / h
r / r’ = (H + h) / h
rh = (H + h)r’
rh = Hr’ + hr’
(r -r’)h = Hr’
h = Hr’ / (r -r’)
Substituting these values in equation (2),
V = 1/3π[{r3h – r3h} / r’]
= 1/3π[{r3 – r’3}h / r’]
= 1/3π[{r3 – r’3}{Hr’ / (r – r’)} / r’]
= 1/3πH(r2 + r’2 +rr’)
Thus,
Volume of the frustum of cone = 1/3 πH(r2 + r’2 + rr’)
Frustum of Cone
Frustum of a cone is a special shape that is formed when we cut the cone with a plane parallel to its base. The cone is a three-dimensional shape having a circular base and a vertex. So the frustum of a cone is a solid volume that is formed by removing a part of the cone with a plane parallel to circular base. The frustum is not only defined for cones but can be also defined for the different types of pyramids (square pyramid, triangular pyramid, etc.).
Some of the common shapes of a frustum of cone which we discover in our daily life are buckets, lamp shade, and others. Let us learn more about the frustum of cones in this article.