Question

Find the surface area of a sphere if its volume is $36\pi $ cubic units.
(a) $3\pi $
(b) $9\pi $
(c) $27\pi $
(d) $36\pi $

Answer 1

Hint: Use the fact that the volume of the sphere with radius ‘r’ is $\dfrac{4}{3}\pi {{r}^{3}}$. Equate the formula to the volume given in the question and simplify the equation to calculate the radius of the sphere. Use the fact that the surface area of the sphere is $4\pi {{r}^{2}}$. Substitute the value of radius and simplify to calculate the surface area of the sphere.

Complete step-by-step solution -
We have to calculate the surface area of the sphere whose volume is $36\pi $ cubic units.
Let’s assume that the radius of this sphere is ‘r’.

We know that the volume of the sphere with radius ‘r’ is $\dfrac{4}{3}\pi {{r}^{3}}$.
Thus, we have $\dfrac{4}{3}\pi {{r}^{3}}=36\pi $.
Simplifying the above equation by rearranging the terms, we have ${{r}^{3}}=\dfrac{36\pi \times 3}{4\pi }$.
Thus, we have ${{r}^{3}}=\dfrac{36\pi \times 3}{4\pi }=27$.
Taking the cube root on both sides, we have $r={{\left( 27 \right)}^{\dfrac{1}{3}}}=3$ units.
We will now calculate the surface area of the sphere.
We know that the surface area of the sphere with radius ‘r’ is $4\pi {{r}^{2}}$.
Substituting $r=3$ in the above formula, the surface area of the sphere is $=4\pi {{\left( 3 \right)}^{2}}=36\pi $ square units.
Hence, the surface area of the sphere is $36\pi $ square units, which is option (d).

Note: We must be careful about the units while calculating the surface area of units. As the volume of the sphere is in cubic units, the radius of the sphere is in units of length and thus, the surface area of the sphere will be in square units.