 UPSKILL MATH PLUS

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### 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 $$R$$ and $$r$$ be the radii of the bases $$(R > r)$$, $$h$$ 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

$$=$$ $$\frac{1}{2}(2 \pi R + 2 \pi r) l$$

$$=$$ $$\frac{1}{2} \times 2 \pi (R + r) l$$

$$=$$ $$\pi (R + r) l$$
Curved surface area of a frustum of a cone $$=$$ $$\pi (R + r) l$$, where $$l = \sqrt{h^2 + (R - r)^2}$$ sq. units
Total surface area:
T. S. A. $$=$$ Curved surface area $$+$$ Area of the bottom circular region $$+$$ Area of the top circular region

$$=$$ $$\pi (R + r) l$$ $$+$$ $$\pi R^2 + \pi r^2$$
Total surface area of a frustum of a cone $$=$$ $$\pi l(R + r)$$ $$+$$ $$\pi R^2 + \pi r^2$$, where $$l = \sqrt{h^2 + (R - r)^2}$$ sq. units
Reference:
Image by Clker-Free-Vector-Images from Pixabay