Derive the formula for the volume of the frustum of cone ,
Answers
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.
Frustum
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 ).
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