Derive the formula for the curved surface area and total surface area of the frustum of a cone.

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 △ABG and △ADF, DF || BG Therefore, △ABG ≅ △ADF DF/BG = AF/AG = AD/AB r2/r1 = (h1 - h)/h1 = (l1 - l)/l1 r2/r1 = 1 - h/h1 = 1 - l/l1 1 - l/l1 = r2/r1 1 - l/l1 = 1 - r2/r1 = r1 - r2/r1 l1/l = r1/(r1 - r2) l1 = r1l/(r1 - r2) CSA of frustum DECB = CSA of cone ABC - CSA cone ADE = πr1l1 - πr2(l1 - l) = πr1(r1l/(r1 - r2)) - πr2(r1l/(r1 - r2) - l) πr1l/(r1 - r2) - πr2(r1l - l(r1 - r2))/(r1 - r2) = πr1²l/(r1 - r2) - πr2(r1l - r1l + r2l)/(r1 - r2) = πr1²l/(r1 - r2) - πr2²l/(r1 - r2) = πl[r1² - r2²]/(r1 - r2) 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. = π(r1 + r2)l + πr2² + πr1² = π[(r1 + r2)l + r1² + r2²]