Question

Find the surface area and total surface area of the hemisphere of radius 21 cm.
(a) $2772\,c{{m}^{2}}$; $4158\,c{{m}^{2}}$
(b) $2442\,c{{m}^{2}}$; $4158\,c{{m}^{2}}$
(c) $2772\,c{{m}^{2}}$; $4874\,c{{m}^{2}}$
(d) $2112\,c{{m}^{2}}$; $4158\,c{{m}^{2}}$

Answer 1

Hint: In this question, first draw the diagram with the given conditions or measurements. Use the formula of surface area and the total surface area of the hemisphere and calculate for the following. Write the final conclusive statement.

Complete step-by-step answer:
Let us draw the figure of the hemisphere of radius 21 cm.

In this question, we have a hemisphere of radius 21 cm. We have to find the surface area and the total surface area.
Let us first find the surface area of the hemisphere.
We know,
Surface area = $2\pi {{r}^{2}}$
 $\begin{align}
  & =2\times \pi \times {{\left( 21 \right)}^{2}} \\
 & =2\times \pi \times 441 \\
 & =882\times \pi \\
 & =2772\\
\end{align}$
Therefore, we can say that the surface area is equal to 2772 sq. cm.
Now, let us find the total surface area of the hemisphere.
If you see, we have two surface areas in hemisphere,
(i) The surface area or the curved surface area.
(ii) The top of the hemisphere where the surface is in the shape of a circle.
So, we can say that,
Total surface area of hemisphere = curved surface area + surface area in the shape of the circle
= $2\pi {{r}^{2}}$ + $\pi {{r}^{2}}$
= $3\pi {{r}^{2}}$
Therefore, total surface area of the hemisphere = $3\pi {{r}^{2}}$
 $\begin{align}
  & =3\times \pi \times {{\left( 21 \right)}^{2}} \\
 & =3\times \pi \times \left( 441 \right) \\
 & =1323\times \pi \\
 & =4158 \\
\end{align}$
Therefore, we can say that the total surface area of the hemisphere is equal to 4158 sq. cm.
Hence, the answer is option (A) that is, the surface area and the total surface area of the hemisphere is $2772\,c{{m}^{2}}$ and $4158\,c{{m}^{2}}$ respectively.

Note: Hemisphere is basically a sphere cut off in the middle either horizontally or vertically. The surface area of the hemisphere is also known as curved surface area. The approximation of the answer was taken to match the answers in the multiple choice which was relatively closer to the obtained result.