Question

dirivation of formula of frustum of cone​

Answer 1

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

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 the volume of the total cone = 1/3 π R2 H’ = 1/3 π R2 (H+h)

The volume of the Tip cone = 1/3 πr2h. For finding the volume of the frustum we calculate the difference between the two right circular cones, this gives us

= 1/3 π R2 H’ -1/3 πr2h
= 1/3π R2 (H+h) -1/3 πr2h
=1/3 π [ R2 (H+h)-r2 h ]

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. This gives us, R/ r = H+h / h ⇒ H+h = Rh/r . Substituting the value of H+h in the formula for the volume of frustum we get,

=1/3 π [ R2 (Rh/r)-r2 h ] =1/3 π  [R3h/r-r2 h )]
=1/3 π h (R3/r-r2 )  =1/3 π h (R3-r3 / r)

The Volume of Frustum of Cone = 1/3 π h [(R3-r3)/ r]

Similar Triangles Property

Using the same Similar triangles property lets find the value of h, R/ r = (H+h)/ h.
⇒ h= [r/(R-r)] H. Substituting the value of h this equation we get: =1/3 πH [r/(R-r)][(R3-r3)/ r)\]
=1/3 πH [(R3-r3)/(R-r)]
= 1/πH [(R-r)(R2 +Rr+r2 )/ (R-r) ]
=1/πH (R2 +Rr+r2 ).
Therefore, the volume (V) of the frustum of the cone is =1/3 πH (R2 +Rr+r2 ).

Learn more about Area and Volume of Combination of Solids here

Curved Surface Area and Total Surface Area of the Frustum

The curved surface area of the frustum of the cone = π(R+r)l1 

The total surface area of the frustum of the cone = π l1 (R+r) +πR2 +πr2

The slant height (l1) in both the cases shall be  = √[H2 +(R-r)2]

These equations have been derived using the similarity of triangles property between the two triangles QPS and QAB. Measurement of volume, surface area and curved surface area like any other measurement depends on the understanding of the subject.

For the combination of solids, we add all the constituting shapes, here since we are slicing a similar triangle from the cone, we find the difference between the two shapes. These two parts of the measurements involve operations and depend highly on the logic of understanding.