### Theory:

Let us look at our surroundings for a moment. We see glass tumblers, buckets, traffic cones in our day to day life. Do you know what shape is these all? If your answer is a cone, then it is wrong.

The shape resembling in the above pictures is called the frustum of a cone.
If a smaller end of the cone is sliced by a plane parallel to its base, the portion of a solid between this plane and the base is known as the frustum of a cone.
Surface area of a frustum of a cone

Let $$r_1$$ and $$r_2$$ be the radii of base $$(r_1 > r_2)$$, $$h$$ be the height, and $$l$$ be the slant height of the frustum of a cone.

Curved surface area:

C. S. A. $$=$$ $$\frac{1}{2}$$ (sum of the circumferences of base and top region) $$\times$$ slant height

$$=$$ $$\frac{1}{2}(2 \pi r_1 + 2 \pi r_2) l$$

$$=$$ $$\frac{1}{2} \times 2 \pi (r_1 + r_2) l$$

$$=$$ $$\pi (r_1 + r_2) l$$
Curved surface area of a frustum of a cone $$=$$ $$\pi (r_1 + r_2) l$$, where $$l = \sqrt{h^2 + (r_1 - r_2)^2}$$ sq. units
Total surface area:

T. S. A. $$=$$ Curved surface area $$+$$ Area of the bottom circular region $$+$$ Area of the top circular region

$$=$$ $$\pi (r_1 + r_2) l$$ $$+$$ $$\pi r_1^2 + \pi r_2^2$$
Total surface area of a frustum of a cone $$=$$ $$\pi l(r_1 + r_2)$$ $$+$$ $$\pi r_1^2 + \pi r_2^2$$, where $$l = \sqrt{h^2 + (r_1 - r_2)^2}$$ sq. units
Volume of a frustum of a cone

Let $$r_1$$ and $$r_2$$ be the radii of base $$(r_1 > r_2)$$, $$h$$ be the height, and $$l$$ be the slant height of the frustum of a cone.

We need to find the height of the smaller cone $$ADE$$.

Consider $$\Delta ABC$$ and $$\Delta ADE$$.

$$\angle BAC = \angle DAE$$ [common angle]

$$\angle ABC = \angle ADE$$ [Both $$90^\circ$$]

Therefore, $$\Delta ABC \sim \Delta ADE$$ [by AA similarity].

The corresponding sides of similar triangles are proportional.

$$\frac{AB}{AD} = \frac{BC}{DE}$$

$$\frac{H - h}{H} = \frac{r_2}{r_1}$$

$$Hr_1 - hr_1 = Hr_2$$

$$Hr_1 - Hr_2 = hr_1$$

$$H(r_1 - r_2) = hr_1$$

$$H = \frac{hr_1}{(r_1 - r_2)}$$ - - - - - - (I)

Volume of frustum $$=$$ Volume of big cone $$-$$ Volume of small cone

$$=$$ Volume of $$ADE$$ cone $$-$$ Volume of $$ABC$$ cone

$$=$$ $$\frac{1}{3} \pi r_1^2H$$ $$-$$ $$\frac{1}{3} \pi r_2^2(H - h)$$

$$=$$ $$\frac{1}{3} \pi r_1^2H$$ $$-$$ $$\frac{1}{3} \pi r_2^2H$$ $$+$$ $$\frac{1}{3} \pi r_2^2h$$

$$=$$ $$\frac{1}{3} \pi H (r_1^2 - r_2^2)$$ $$+$$ $$\frac{1}{3} \pi r_2^2h$$

$$=$$ $$\frac{1}{3} \pi \times \frac{hr_1}{(r_1 - r_2)} (r_1^2 - r_2^2)$$ $$+$$ $$\frac{1}{3} \pi r_2^2h$$ [using equation (I)]

$$=$$ $$\frac{1}{3} \pi \times \frac{hr_1}{(r_1 - r_2)} (r_1 - r_2) (r_1 + r_2)$$ $$+$$ $$\frac{1}{3} \pi r_2^2h$$

$$=$$ $$\frac{1}{3} \pi hr_1(r_1 + r_2)$$ $$+$$ $$\frac{1}{3} \pi r_2^2h$$

$$=$$ $$\frac{1}{3} \pi h[r_1(r_1 + r_2) + r_2^2]$$

$$=$$ $$\frac{1}{3} \pi h[r_1^2 + r_1r_2 + r_2^2]$$
Volume of the frustum of a cone $$=$$ $$\frac{1}{3} \pi h[r_1^2 + r_1r_2 + r_2^2]$$ cu. units
Reference:
Image by Clker-Free-Vector-Images from Pixabay