a sector of a circle radius 9cm and central angle of 120 it is rolled up so that the two bounding radii joined together to form a cone find the slant height of the cone, radius of the base of the cone, volume of the cone , and the  T.S.A of the cone?

Dear Student,

Please find below the solution to the asked query:

Given : A sector of a circle radius 9 cm and central angle of 120$°$ , It is rolled up so that the two bounding radii joined together to form a cone .

So, Circumference of base of cone = Length of the arc of sector

We know length of arc of a sector  = $\left(\frac{\theta }{360°}\right)\times 2\times \mathrm{\pi }\times \mathrm{r}$ , So

Length of the arc of given sector = $\left(\frac{120°}{360°}\right)\times 2\times \mathrm{\pi }\times 9 = \left(\frac{1}{3}\right)\times 2\times \mathrm{\pi }\times 9 = 2\times \mathrm{\pi }\times 3 = \mathbf{6}\mathbf{ }\mathbf{\pi }\mathbf{ }\mathbf{ }\mathbf{cm}$

We know circumference of circle = $2\mathrm{\pi r}$ , So

Circumference of base of cone = $2\mathrm{\pi r}$ , Then

$$\begin{gathered} 2\mathrm{\pi r} = 6\mathrm{\pi } \\ \mathbf{\Rightarrow }\mathit{r}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{3} \end{gathered}$$
So,
Radius of base of cone = r  =  3 cm                                                                  ( Ans )

Slant height of cone  l =  Radius of sector  = 9 cm                                               ( Ans ) 

we know :  Slant height = l = $\sqrt{h^2 + r^2}$ , Here , h  =  Height of cone and r  = Radius of cone   , So
$$\begin{gathered} \sqrt{h^2 +3^2} = 9 \\ \Rightarrow h^2 + 3^2 = 9^2 \\ \Rightarrow h^2 +9 = 81 \\ \Rightarrow h^2 = 72 \\ \Rightarrow h =\sqrt{ 72} \\ \Rightarrow h =6\sqrt{2} \\ \Rightarrow h =6 \times 1.414 \\ \Rightarrow h =8.484 \approx \mathbf{8}\mathbf{.}\mathbf{48}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{cm} \end{gathered}$$
Therefore,

Height of cone  = h  = 8.48 cm                                                        ( Ans )

We know volume of cone  = $\frac{\mathrm{\pi }\mathit{ }{r}^{\mathit{2}}\mathit{ }h}{3}$ , So

Volume of given cone = $\frac{{\displaystyle \frac{22}{7}}\times \mathit{ }{3}^{\mathit{2}}\mathit{ }\times 8.48}{3} = \frac{{\displaystyle 22\times \mathit{ }3\times 8.48}}{7}= \frac{{\displaystyle 559.68}}{7} =\mathbf{79}\mathbf{.}\mathbf{95}\mathbf{ }{\mathbf{cm}}^{\mathbf{3}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)}$

And

We know total surface area of cone  = $\mathrm{\pi }\mathit{ }r\left( r +\hspace{0.17em}l \right)$ , So

Total surface area of cone = $\frac{22}{7}\times 3\left( 3 +9 \right) = \frac{22}{7}\times 3\times 12 = \frac{792}{7} = \mathbf{113}\mathbf{.}\mathbf{14}\mathbf{ }{\mathbf{cm}}^{\mathbf{2}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)}$

Hope this information will clear your doubts about Surface Areas and Volumes.

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