Volume of a shape is defined as how much capacity a shape has or we can say how much material was required to form that shape. A hemisphere, derived from the Greek words "hemi" (meaning half) and "sphere," is simply half of a sphere. If you imagine slicing a perfectly round sphere into two equal halves, each half would be a hemisphere.
To calculate the volume of a hemisphere, you can use the formula derived from the volume of a sphere [(4/3)πr3]. Since a hemisphere is half of a sphere, its volume is half of the sphere’s volume [(2/3)πr3].
In this article, we will discuss the volume of a hemisphere in detail, including its formula as well as solved examples.
Table of Content
What is Hemisphere?
Hemisphere can be defined as a 3-dimensional shape that is formed by cutting a sphere into two equal halves. It is a combination of a half-spherical curve and a plane circular region.
Some common examples of hemispheres are:
- Northern Hemisphere and Southern Hemisphere of Earth
- Bowls in Kitchen
- Domes in Architecture
- Half of Orange and Watermelon
Volume of Hemisphere Formula
Hemisphere is just half of a sphere so its volume will also be just half. As we know the volume of a sphere is 4/3πr3, thus formula for volume of hemisphere is:
Volume of Hemisphere = Volume of sphere/2 = (2/3)πr3
Derivation of Volume of Hemisphere
It has been experimentally proved that the volume of a sphere is 2/3 of the volume of a cylinder with the same radius, and height equal to the diameter.
Volume of a cylinder with radius r and height as 2r = πr2(2r) = 2πr3
So, the volume of the sphere will be = 2/3 × (2πr3) = 4/3πr3
And similarly, the volume of the hemisphere can also be derived by dividing the volume of the sphere by 2.
Hence,
Volume of Hemisphere = 2/3πr3
Where r is the radius of Hemisphere.
How to Find the Volume of a Hemisphere?
Volume of a hemisphere is calculated using the formula, Volume = 2πr3/3.
Use the following steps for finding the volume of a hemisphere.
Example: Find the volume of the hemisphere with a radius of 14 cm
Solution:
Step 1: Find the radius of a hemisphere, radius (r) = 14 cm
Step 2: Put the value of the radius in the given formula, Volume of hemisphere = 2πr3/3
Step 3: Solving the formula for volume,Volume of hemisphere = 2πr3/3
= (2 × 3.14 × 143)/3
= 5744.10667 cm3
Volume of Hollow Hemisphere
A hollow hemisphere is a three-dimensional shape with an inner and outer sphere. It resembles a dome with a certain thickness, which is the difference between the radii of the inner and outer spheres.
The formula for the volume of a hollow hemisphere is:
Volume of Hollow Hemisphere = (2/3) π (R3 − r3)
Where,
- R is the radius of the outer sphere, and
- r is the radius of the inner sphere.
Read More,
- Mensuration
- Surface Area of a Hemisphere
- Volume of Sphere
- Volume of Cone
- Volume of Cube
- Volume of Cylinder
Solved Examples on Volume of Hemisphere
Example 1: If the radius of a hemisphere is 21 cm. Find the volume of the hemisphere.
Solution:
Given: r = 21 cm
We know that volume of hemisphere = 2/3 π r3
Volume of Hemisphere = 2/3 x 22/7 x 21 x 21 x 21
⇒ Volume of Hemisphere = 2 x 22 x 21 x 21
⇒ Volume of Hemisphere = 19404 cm3
Example 2: If the volume of the hemisphere is 30 cubic meters. It is melted and used to form hemispheres with a volume of 10 cubic meters. How many such hemispheres can be made?
Solution:
Certain number of hemispheres are made using a single big hemisphere.
Volume of small hemispheres = Volume of large hemisphere
Let there be n number of small hemispheres .
Then, n x volume of small hemisphere = volume of large hemisphere
⇒ n x 10 = 30
⇒ n = 30/10
⇒ n = 3
So, three hemispheres can be formed by melting the bigger hemisphere.
Example 3: Find the volume of a hemisphere of diameter 5 cm.
Solution:
Given: Diameter = 5 cm
Thus, r = 5/2 [Diameter = 2 (Radius)
Volume of Hemisphere = 2/3 π r3
⇒ Volume = 2/3 π (5/2)3
⇒ Volume = 32.724 cm3
Example 4: If a hemisphere of radius 2 cm is fitted inside a cuboid and then water is filled inside the cuboid. Find the amount of water present in the cuboid.
Solution:
Length of cuboid = 2r = 4 cm
Breadth of cuboid = 2r = 4 cm
Height of cuboid = r = 2 cm
Volume of Cuboid = lbh = 2 x 4 x 4 = 32 cm3
Volume of Hemisphere = 2/3 π r3
⇒ Volume of Hemisphere = 2/3 x (3.14) x (2)3
⇒ Volume of Hemisphere = 16.75 cm3
Thus, Volume of water = volume of cuboid - volume of hemisphere
⇒ Volume of Water = 32 - 16.75
⇒ Volume of Water = 15.25 cm3
So, amount of water present in cuboid is 15.25 cm3
Example 5: If the volume of the hemisphere is 2.095 m3. Find the radius of the hemisphere.
Solution:
Volume of hemisphere = 2/3 π r3
2.095 = 2/3 π r3
2.095 = 2.095 r3
r3 = 1
r = 1 m
So, radius of hemisphere is 1 m.
Practice Problems on Volume of Hemisphere
Problem 1: Find the volume of a hemisphere with a radius of 10 cm. Use π=3.14.
Problem 2: A hemispherical bowl has a diameter of 12 inches. Calculate its volume.
Problem 3: Determine the volume of a hemisphere whose radius is 7 meters.
Problem 4: The radius of a hollow hemisphere is 15 cm, and the thickness of the material is 3 cm. Calculate the volume of the hollow part.
Problem 5: If a sphere with a radius of 8 cm is cut into two equal hemispheres, what is the volume of each hemisphere?