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Question
Derive the formula for the curved surface area and total surface area of the frustum of 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 - 1/l_1`
`l - l/l_1= r_2/r_1`
`l/l_1 =1-r_2/r_1 =(r_1-r_2)/r_1`
`l_1/l = r_1/(r_1-r_2)`
`l_1 = (r_1l)/(r_1-r_2)`
CSA of frustum DECB = CSA of cone ABC − CSA cone ADE
`= pir_1l_1 - pir_2(l_1-l)`
`=pir_1((lr_1)/(r_1-r_2))-pir_2[(r_1l)/(r_1-r_2)-l]`
`= (pir_1^2l)/(r_1-r_2) - pir_2((r_1l-r_1l+r_2l)/(r_1-r_2))`
`=(pir_1^2l)/(r_1-r_2)-(pir_2^2l)/(r_1-r_2)`
`= pil[(r_1^2-r_2^2)/(r_1-r_2)]`
CSA of frustum = Π(r1 + r2)l
Total surface area of frustum = CSA of frustum + Area of upper circular end + Area of lower circular end
`= pi(r_1+r_2)l+pir_2^2+pir_1^2`
`=pi[(r_1+r_2)l+r_1^2+r_2^2]`
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