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# Derive the formula for the curved surface area and total surface area of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.

Let ABC be a cone. From the cone the frustum DECB is cut by a plane parallel to its base. Here, r1 and r2 are the radii of the frustum ends of the cone and h be the frustum height.

Now, consider the $$\triangle$$ ABG $$\triangle$$ ADF,

Here, DF||BG

So, $$\triangle \ ABG \ \sim \ \triangle \ ADF$$

$$\Rightarrow \frac{DF}{BG} \ = \ \frac{AF}{AG} \ = \ \frac{AD}{AB}$$

$$\Rightarrow \frac{r_2}{r_1} \ = \ \frac{h_1 \ - \ h}{h_1} \ = \ \frac{l_1 \ - \ l}{l_1}$$

$$\Rightarrow \ l_1 \ = \ \frac{r_1 l}{r_1 \ - \ r_2}$$

The total surface area of frustum will be equal to the total CSA ot frustum + the area ol upper circular end + area of the lower circular end

$$= \pi (r_1 \ + \ r_2)l \ + \ \pi (r_2)^2 \ + \ \pi (r_1)^2$$

$$\therefore$$ Surface area of frustum $$= \ \pi [r_1 \ + \ r_2)l \ + \ (r_1)^2 \ + \ (r_2)^2]$$