Question

Write the formula of curved and total surface area of a hemisphere. Also find the ratio between them.

Answer 1

Hint: We use the curved surface area of a sphere and the concept that hemisphere is half of the sphere. Use the concept that the base of the hemisphere is a circle.
* Curved surface area of any figure is the surface area of the curved part of any figure.
* Total surface area of any figure is the curved surface area plus the flat surface area of the figure.
* Curved surface area of the sphere is \[4\pi {r^2}\].
* Ratio is comparison of two values with each other which is given by \[\dfrac{a}{b} = a:b\], where a and b have no common factor.

Complete step-by-step answer:
We know a hemisphere is the exact half part of the sphere.
So, the curved surface area of the hemisphere will be exactly half the value of the curved surface area of the sphere.
\[ \Rightarrow \]Curved surface area of hemisphere\[ = \dfrac{1}{2}\left( {4\pi {r^2}} \right)\]
Cancel same factors from numerator and denominator in the formula
\[ \Rightarrow \]Curved surface area of hemisphere\[ = 2\pi {r^2}\]....................… (1)
Now we have to find the total surface area of the hemisphere.
Since the base of the hemisphere is a circle.
A circle having radius ‘r’\[ = \pi {r^2}\].................… (2)
Total surface area of any figure is the curved surface area plus the flat surface area of the figure.
Add the area in equation (1) and equation (2)
\[ \Rightarrow \]Total surface area of hemisphere\[ = 2\pi {r^2} + \pi {r^2}\]
Take \[\pi {r^2}\]common
\[ \Rightarrow \]Total surface area of hemisphere\[ = \pi {r^2}(2 + 1)\]
\[ \Rightarrow \]Total surface area of hemisphere\[ = 3\pi {r^2}\]...................… (3)
\[\therefore \]Curved surface area of hemisphere is \[2\pi {r^2}\] square units and Total surface area of hemisphere is \[3\pi {r^2}\]square units.
Now the ratio of Curved surface area to the total surface area is given by writing curved surface area as numerator and total surface area as denominator of the fraction.
From equation (1) and (3), the ratio of curved surface area : total surface area is \[\dfrac{{2\pi {r^2}}}{{3\pi {r^2}}}\]
Cancel same factors from numerator and denominator
\[ \Rightarrow \]The ratio of curved surface area : total surface area is \[\dfrac{2}{3}\]

\[\therefore \]The ratio of curved surface area : total surface area is \[2:3\]

Note: Students mostly make the mistake of writing the ratio as \[2\pi {r^2}:3\pi {r^2}\], which is wrong because the ratio should always be in simplest form, there should be no common factor between the numerator and denominator.