Question

Find the curved surface area (C.S.A.) and total surface area (T.S.A.) of a solid hemisphere of radius $ 14\,\,cm $ .

Answer 1

Hint: To find curved surface area and total surface area of a hemisphere we first write their fixed mensuration formulas. And then substituting values of given radius in two different formulas and simplifying them to get the required solution of the problem.
Curved surface area (C.S.A.) of a hemi sphere is given as: $ 2\pi {r^2} $ .
Total surface area (T.S.A.) of a hemi sphere is given as: $ 3\pi {r^2} $

Complete step-by-step answer:
The radius of a hemisphere is $ 14\,\,cm $ .
We know that hemisphere is part of a sphere.

If we divide a given sphere in two equal parts a hemi sphere is obtained.
And hence from above we can say that the curved surface area of a hemisphere will be half of a sphere from which it is obtained.
Surface area of a sphere = $ 4\pi {r^2} $
Curved surface area of a hemisphere = $ \dfrac{1}{2} $ of surface area of a sphere.
 $ \Rightarrow $ Curved surface area of a hemisphere = $ \dfrac{1}{2}\left( {4\pi {r^2}} \right) $
Substituting value of $ r = 14\,cm $ in above formulas. We have,
Curved surface area of a hemisphere = $ \dfrac{1}{2} \times 4 \times \dfrac{{22}}{7} \times 14 \times 14 $
\[
\Rightarrow \,Curved\,\,surface\,\,area\,\,of\,\,a\,\,hemi\,\,sphere\,\, = 2 \times 22 \times 2 \times 14 \\
 \Rightarrow Curved\,\,surface\,\,area\,\,of\,\,a\,\,hemisphere\,\, = 44 \times 28 \\
  \Rightarrow Curved\,\,surface\,\,area\,\,of\,\,a\,\,hemisphere\,\, = 1232 \;
 \]
Therefore, from above we see that the curved surface area of a hemi sphere of radius $ 14\,\,cm $ is $ 1232\,\,c{m^2} $ .
Also, we know that when a sphere is divided in two parts a hemi sphere is formed.
A hemisphere formed has a curved surface area but there is one more surface formed on the upper part of the sphere where a sphere is cut a plane.
Hence, the total surface area of a hemisphere will be a sum of curved surface area and area of circle formed on the upper part of the hemisphere when a sphere is cut into two parts.
 $ \therefore $ Total surface area (T.S.A.) of a hemisphere = Curved surface area (C.S.A.) + Area of a circle formed.
 $ \Rightarrow $ Total surface area of a hemisphere = $ 2\pi {r^2} + \pi {r^2} $
 $ \Rightarrow $ Total surface area of a hemisphere = $ 3\pi {r^2} $
Substituting value of radius r = $ 14\,\,cm $ in above equation. We have,
$
Total\,\,sufrace\,\,area(T.S.A.)\,\,of\,\,a\,\,hemisphere\,\, = \,3 \times \dfrac{{22}}{7} \times 14 \times 14 \\
\Rightarrow Total\,\,sufrace\,\,area(T.S.A.)\,\,of\,\,a\,\,hemisphere\,\, = \,3 \times 22 \times 14 \times 2 \\
\Rightarrow Total\,\,sufrace\,\,area(T.S.A.)\,\,of\,\,a\,\,hemisphere\,\, = \,66 \times 28 \\
 \Rightarrow Total\,\,sufrace\,\,area(T.S.A.)\,\,of\,\,a\,\,hemisphere\,\, = \,\,1848 \\
  $
Therefore, from above we see that the total surface area of a hemisphere of radius $ 14\,\,cm $ is $ 1848\,\,c{m^2} $ .

Note: In case of any mensuration problem one must see units of terms given in a problem. If there are any terms with different units then the first step makes units of all terms the same and then choose a mensuration formulas related to the given figure and finally simplifying carefully after substituting values from the given problem to get the required solution of the problem.