A cone of radius 90 cm and height 120 cm stands on its base. It is cut into 3 parts by 2 cuts parallel to its base such that the heights of the three parts (from top to bottom) are in ratio of 1 : 2 : 3. What is the total surface area (in cm²) of the middle part?

Because the heights of the three parts from top to bottom are in a 1 to 2 to 3 ratio, the middle part spans from a height of 20 cm from the top to 60 cm from the top. By similar triangles, the top radius at 20 cm is 15 cm and at 60 cm is 45 cm, making the slant height of the middle frustum the square root of ((45-15)^2 + 40^2), which is 50 cm. The total surface area of the middle part is the sum of the lateral surface area and the two base areas, calculated as pi*(45+15)50 + pi(45^2) + pi*(15^2) = 3000*pi + 2025*pi + 225*pi = 5250*pi, which equals 16500 cm2.

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