What is the Volume of a Hemisphere?
The word ‘hemisphere’ is originally derived from the Greek word ‘hemi’, which means half and the word ‘sphaira’, which refers to the term ‘globe’. As the name suggests, a hemisphere is half of a sphere. The volume of a sphere is equal to the sum of the volumes of two hemispheres. A hemisphere has one curved surface and one flat circular base. The volume of any object is measured in cubic units, so, a hemisphere’s volume is expressed in cubic units such as m3, cm3, in3 and other cubic units.
How to Calculate the Volume of a Hemisphere?
The formula for the volume of a hemisphere is derived by dividing the volume of the sphere by two.
Volume of Hemisphere = \(\frac{\text{Volume of sphere}}{2}\)
By substituting the formula for the volume of a sphere,
Volume of Hemisphere = \(\frac{\frac{4}{3}\pi~r^3}{2}\)
Or
Volume of Hemisphere = \(\frac{2}{3}\pi~r^3\)
Here, ‘r’ indicates the radius of the hemisphere. A quick tip to memorize the volume of a hemisphere formula is to simply remember the volume of a sphere formula and divide it by 2.
Properties of a Hemisphere
The properties of a hemisphere are listed below:
- It has a curved surface area.
- It has a curved edge where both the curved face and the flat face meet.
- The circular base of the hemisphere acts as a flat base.
- A hemisphere has no vertices.
- A hemisphere is not a polyhedron as polyhedrons are formed by polygons as units.
Rapid Recall
Solved Examples
Example 1:
If a sphere of radius 6 meter is cut into two equal parts, what would be the volume of each obtained hemisphere?
Solution:
The radius of the sphere = 6 m
The same sphere is sliced into two equal halves, the radius of the hemispheres that are formed will also be 6 m.
Then, the volume of the hemispheres will be,
\(V=\frac{2}{3}\pi~r^3\) [Write the formula for volume of sphere]
\(~=\frac{2}{3}\times~3.14~\times~(6)^3\) [Substitute the radius value]
\(~=\frac{2}{3}\times~3.14~\times~216\) [Multiply]
\(~=\frac{2}{3}\times~678.24\) [Multiply]
\(~=\frac{1356.48}{3}\) [Simplify]
\(~=452.16~m^3\) [Divide]
Therefore, the volume of the hemisphere is 452.16 cubic meters.
Example 2:
Richard has bought a watermelon, which has a diameter of 16 centimeters. He cut the watermelon in two so that he and his brother Rick could each have one half of the watermelon. Find the volume of the watermelon portion received by Rick.(Solve the problem by assuming the shape of watermelon as a sphere)
Solution:
The diameter of the watermelon = 16 cm
The radius = \(\frac{16}{2}=8~cm\)
Here, a watermelon is assumed to be spherical in shape.
So, the volume of one half of the watermelon is calculated by using the formula for the volume of a hemisphere.
\(V=\frac{2}{3}\pi~r^3\) [Write the formula for volume of hemisphere]
\(~=\frac{2}{3}\times~3.14~\times~(8)^3\) [Substitute the radius value]
\(~=\frac{2}{3}\times~3.14~\times~512\) [Multiply]
\(~=\frac{2}{3}\times~1607.68\) [Multiply]
\(~=\frac{3215.36}{3}\) [Multiply]
\(\approx ~1071.78\) [Divide]
Hence, the volume of Rick’s portion of the watermelon is 1071.78cubic centimeters.
Example 3:
Jenny’s mother asked her to cut the oranges into two pieces so she can bake an orange pie. While cutting one of the oranges, Jenny noticed that the shape of the orange is a hemisphere and wondered what would be its radius. Can you help Jenny in finding the radius of the orange if its volume is 218 cubic centimeters.
Solution:
The volume of the orange = \(218~cm^3\).
From this, the volume of the resultant hemispheres after cutting the orange could be calculated by the formula:
Volume of Hemisphere =\(\frac{\text{Volume of sphere}}{2}\)
⇒ \(\frac{218}{2}\) [Substitute the given volume]
⇒ \(109~cm^3\) [Divide]
Use the formula for the volume of a hemisphere to find the volume of half an orange.
\(V=\frac{2}{3}\pi~r^3\) [Write the formula]
\(109=\frac{2}{3}~\times~3.14\times~r^3\) [Substitute the values]
\(327=6.28~\times~r^3\) [Multiply each side by 3]
\(\frac{327}{6.28}=r^3\) [Divide each side by 6.28]
\(52.07=r^3\) [Simplify]
\(\sqrt[3]{52.07}=r\) [apply cube rooting on both sides]
\(3.734~\cong~r\) [Simplify]
Therefore, the radius of the orange slice is approximately 3.734 centimeters.
An igloo, a bowl, cups, ice-cream scoops, the hemisphere of the Earth, mushrooms, and a service bell, are a few real-life examples of hemispheres.
Yes, by multiplying the volume of a hemisphere by two, we can calculate the volume of a sphere.
A hemisphere only has two surfaces, one is flat and circular and a dome-shaped portion known as the curved surface.