Question

Find the total surface area of a hemisphere of radius 10 cm. (Use$\pi =3.142$)
A.942 $c{{m}^{2}}$
B.492 $c{{m}^{2}}$
C.249 $c{{m}^{2}}$
D.924 $c{{m}^{2}}$

Answer 1

Hint: Use the formula ‘Total surface area of hemisphere $=3\pi {{r}^{2}}$’ and put the value of radius equal to 10 cm. to get the answer.

Complete step-by-step answer:

To find the total surface area of hemisphere we will first derive the formula of it,
As we know that the surface area of the complete sphere is $4\pi {{r}^{2}}$.
Therefore the surface area of the semicircular part of the sphere is half of the surface area of the sphere.
Therefore,
A (Semicircular Part) $=\dfrac{surface\text{ }area\text{ }of\text{ }complete\text{ }sphere}{2}$
Therefore, A (Semicircular Part) $=\dfrac{4\pi {{r}^{2}}}{2}$
Therefore, A (Semicircular Part) $=2\pi {{r}^{2}}$ ……………………………………… (1)
Also area of circle is given by,
A (Circle) = $\pi {{r}^{2}}$ ……………………………………………………………………………. (2)
Now to derive the total surface area we should know that the semicircle consists of a semicircular part and a circular base and therefore by adding the two areas we can find the total surface area of the Hemisphere.

Therefore,
Total surface area of hemisphere = Surface area of semicircular part + Area of circular base.
Total surface area of hemisphere = A (Semicircular Part) + A (Circle)
If we put the values of equation (1) and equation (2) we will get,
Therefore, Total surface area of hemisphere $=2\pi {{r}^{2}}+\pi {{r}^{2}}$
Therefore, Total surface area of hemisphere $=3\pi {{r}^{2}}$ ……………………………….. (3)
Now we will write the given values,
Radius of hemisphere = r = 410 cm.
$\pi =3.142$
To find the total surface area of the hemisphere we will put the given values in formula (3), therefore we will get,
Therefore, Total surface area of hemisphere $=3\times \left( 3.142 \right)\times {{\left( 10 \right)}^{2}}$
Therefore, Total surface area of hemisphere $=3\times \left( 3.142 \right)\times 100$
Therefore, Total surface area of hemisphere $=3\times 314.2$
Therefore, Total surface area of hemisphere $=942.6$ $c{{m}^{2}}$
Therefore, Total surface area of hemisphere $\approx 942$ $c{{m}^{2}}$
Therefore the total surface area of the hemisphere is equal to 942$c{{m}^{2}}$.
Therefore the correct answer for the above question is option (a).

Note: There are chances of considering the total surface area of hemisphere as $2\pi {{r}^{2}}$by just taking half of the surface area of sphere, but do remember that when we cut the sphere there introduces the circular base which you have to calculate while measuring the total surface area.