Question

Find the cub (cubical) volume of a hemisphere having diameter 1 cm.
A. $\dfrac{\pi }{6}c{m^3}$
B. $\dfrac{\pi }{{12}}c{m^3}$
C. $\dfrac{{2\pi }}{3}c{m^3}$
D. $\dfrac{{4\pi }}{3}c{m^3}$

Answer 1

Hint: We will first write the formula for the area of a hemisphere and then put in the value of radius as mentioned in the question. We will thus have our answer and we just need to match it to some option among given ones.

Complete step-by-step answer:
Let us first get to know what a hemisphere is.
Hemi means half. So hemisphere means half sphere.
Since the sphere is cut into half. Hence, its volume must also be halved.
Since the volume of a sphere is given by $V = \dfrac{4}{3}\pi {r^3}$.
Hence, the volume of the hemisphere will be $V = \dfrac{1}{2} \times \dfrac{4}{3}\pi {r^3} = \dfrac{2}{3}\pi {r^3}$.
Now, we have the formula with us that is $V = \dfrac{2}{3}\pi {r^3}$ ………(1)
Now, we have a hemisphere of diameter 1 cm.
               

We know that: $radius = \dfrac{{diameter}}{2}$.
So, the radius of the hemisphere will be $radius = \dfrac{1}{2}cm$.
Putting this value in (1) will result in:-
$V = \dfrac{2}{3}\pi {r^3} = \dfrac{2}{3} \times \pi \times {\left( {\dfrac{1}{2}} \right)^3}$
Simplifying it, we will have:-
$ \Rightarrow V = \dfrac{2}{3} \times \pi \times \left( {\dfrac{1}{2}} \right) \times \left( {\dfrac{1}{2}} \right) \times \left( {\dfrac{1}{2}} \right)c{m^3}$
$ \Rightarrow V = \dfrac{1}{3} \times \pi \times \dfrac{1}{4}c{m^3}$
$ \Rightarrow V = \dfrac{\pi }{{12}}c{m^3}$
Hence, the answer is $\dfrac{\pi }{{12}}c{m^3}$.

So, the correct answer is “Option B”.

Note: The students must keep in mind to check that if options have $\pi $ in them or not. So, according to that they should put in the values or not.
This question could have been a fill in the blank question as well. So, for that you may solve any way that is either by putting the value of $\pi $ or not and do not forget to write the units of volume at its end. Because writing 40 or 50 does not mean volume or represent volume if it does not have cubic units at its end.
Fun Fact:- In geometry it is an exact half of a sphere. It also refers to half of the Earth, such as the "Northern Hemisphere" (that part of the Earth north of the equator), or "Western Hemisphere" (the half of the Earth west of a line running from the North Pole through England to the South Pole, includes the Americas).