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Find the total surface area of the sphere whose volume is equal to the volume of the cone having the radius 12 cm and height 8 cm.

Last updated date: 14th Jul 2024
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Hint: Find the volume of the cone and then find the radius of the sphere from the volume of cone as they are equal. Substitute the value of radius in the equation of surface area of sphere. Simplify it and find the surface area of the sphere.

Complete step-by-step answer:
We have been given the dimensions of a cone. The radius of the base of the cone is 12 cm and the radius can be represented as ‘r’.
The height of the cone can be marked as ‘h’ which is 6 cm.
We know the volume of a cone is \[{{{}^{1}/{}_{3}}^{rd}}\]of the volume of the cylinder.
Volume of cone =\[\dfrac{1}{3}\pi {{r}^{2}}h\].
It is said that the volume of the cone is equal to the volume of the sphere.
The volume of the sphere is given as\[\dfrac{4}{3}\pi {{r}^{2}}\].
Volume of sphere = Volume of cone……….. (1)
Substitute the volume of sphere and volume of cone and find the radius of the sphere, which can be taken as ‘r’.
\[\dfrac{4}{3}\pi {{r}^{3}}=\dfrac{1}{3}\pi {{r}^{2}}h\].
Substitute the value of radius r = 12 cm and height h = 6 cm in the equation.
\[\dfrac{4}{3}\pi {{r}^{3}}=\dfrac{1}{3}\pi \times {{12}^{2}}\times 6\]. [Cancel out \[\pi \]on LHS and RHS]
\[{{r}^{3}}=\dfrac{3}{4}\times \dfrac{1}{3}\times 12\times 6\times 12\].
\[{{\pi }^{3}}=\dfrac{12\times 12\times 6}{4}=\dfrac{864}{4}=216\].
Take cube root on both LHS and RHS.
\[r=\sqrt[3]{216}=\sqrt[3]{6\times 6\times 6}=6cm\].
Therefore, radius of the sphere, r = 6 cm.
Now what we need to find is the surface area of the sphere.
The surface area of a sphere is given by\[4\pi {{r}^{2}}\], where r = 6 cm.
Surface area of sphere =\[4\pi {{r}^{2}}\].
               =\[4\times \dfrac{22}{7}\times 6\times 6\] [\[\pi \]can be taken as\[{}^{22}/{}_{7}\]]
               = 452.57 sq. cm.
               = 452.57\[c{{m}^{2}}\].
The surface area of sphere = 452.57\[c{{m}^{2}}\].

Note: A sphere is perfectly symmetrical. All the points on the surface are the same distance ‘r’ from the center. The main difference between a sphere and circle is that spheres represent 3 dimensionally, which circles are mentioned in 2 dimensions. But radius is taken the same way.