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Question
Derive the formula for the volume of the frustum of a cone.
Solution
Let ABC be a cone. A frustum DECB is cut by a plane parallel to its base.
Let r1 and r2 be the radii of the ends of the frustum of the cone and h be the height of the frustum of the cone.
In ΔABG and ΔADF, DF||BG
∴ ΔABG ∼ ΔADF
DF/BG = AF/AG = AD/AB
`r_2/r_1 = (h_1-h)/h_1 = (l_1-l)/l_1`
`r_2/r_1 =1 -h/h_1 =1 -l/l_1`
`1-h/h_1= r_2/r_1`
`h/h_1 =1 -r_2/r_1 = (r_1-r_2)/r_1`
`h_1/h= r_1/(r_1-r_2)`
`h_1 = (r_1h)/(r_1-r_2)`
Volume of frustum of cone = Volume of cone ABC − Volume of cone ADE
`=1/3pir_1^2h_1 - 1/3pir_2^2(h_1-h)`
`=pi/3[r_1^2h_1-r_2^2(h_1-h)]`
`=pi/3[r_1^2((hr_1)/(r_1-r_2))-r_2^2((hr_1)/(r_1-r_2)-h)]`
`=pi/3[((hr_1^3)/(r_1-r_2))-r_2^2((hr_1-hr_1+hr_2)/(r_1-r_2))]`
`=pi/3[(hr_1^3)/(r_1-r_2)-(hr_2^3)/(r_1-r_2)]`
`=pi/3h[(r_1^3-r_2^3)/(r_1-r_2)]`
`=pi/3h[((r_1-r_2)(r_1^2+r_2^2+r_1r_2))/(r_1-r_2)]`
`= 1/3pih[r_1^2+r_2^2+r_1r_2]`
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