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.
 
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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.
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Surface area of a frustum of a cone
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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
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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