Volume of Hemisphere

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.

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

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.

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

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

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.

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?

What is a Hemisphere?

A hemisphere is a 3-D shape, it is half of a sphere. Cutting the sphere into two parts results in the creation of two hemispheres.

If a sphere and a hemisphere have the same radii then what is the ratio of their volume?

If a sphere (volume V₁) and a hemisphere(volume V₂) have the same radii then the ratio of their volume is given by

V₁ : V₂ =  (4/3πr³) : (2/3πr³) = 2 : 1

How do we measure the volume of the hemisphere?

The volume of the hemisphere is measured in m³, cm³, liters, etc. The m³ is the standard unit of measurement. We can be converted m³ to liters according to our requirement.

Write the formula for the surface area of the hemisphere.

The surface area of any hemisphere is given by:

• CSA = 2πr²
• TSA = 3πr²

Where ‘r’ is the radius of the given sphere.