Question

The increase in the total surface area of a sphere of radius R when it is cut to make two hemispheres of same radius will be equal to-
$A.\;5\pi {R^2}$
${\text{B}}.\;4\pi {R^2}$
${\text{C}}.\;3\pi {R^2}$
${\text{D}}.\;2\pi {R^2}$

Answer 1

Hint: The surface area of a sphere is given by the formula $4\pi {R^2}$, the curved surface area of the hemisphere is half the value. After calculating the surface areas, we will find their difference to find the answer.

Complete step-by-step solution -

We know that the total surface area of a sphere is given by $4\pi {R^2}$.
The curved surface area of the hemisphere is $2\pi {R^2}$.
The total surface area of the hemisphere is the sum of the curved surface area and the flat surface of radius R. So the total surface area is-
$\begin{gathered}
   = 2\pi {R^2} + \pi {R^2} \\
   = 3\pi {R^2} \\
\end{gathered} $
The total surface area of two such hemispheres is $6\pi {R^2}$.
The increase in the total surface area is $6\pi {R^2}$ - $4\pi {R^2}$ = $2\pi {R^2}$
Hence, the correct option is D.

Note: Students often forget to add the surface area of the circle, that is the flat surface. As a result, they get the incorrect answer. We should know the difference between the total surface area and the curved surface area.