Volume of Hemisphere

Mensuration is the branch of mathematics that deals with different geometric figures and calculating their parameters like length, breadth, radius, volume, surface area, lateral surface area, etc. The 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.

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. 

What is the Volume of a Hemisphere?

A hemisphere is a three-dimensional object. It is half of a sphere. If a sphere is cut by a plane passing through its center, two hemispheres are obtained. A hemisphere has a flat circular base. The volume of a hemisphere is defined as the number of unit cubes it can hold.

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

Volume of Hemisphere = Volume of sphere/2

                                      = (4/3πr³)/2

                                      = 2/3πr³

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

Hemisphere Equation

Suppose, the radius of the is ‘r‘ units and its center is at origin (0,0,0) then its equation is given by,

x² + y² + z² = r²

Cartesian form or equation of a hemisphere with the radius “r” at the center point (x₁, y₁, z₁) is,

(x-x₁)² + (y- y₁)² + (z- z₁)² = r²

Spherical Coordinates of the hemisphere are,

x = r cos θ sin ∅
y = r sin θ cos ∅
z = r cos ∅

Solved Examples on Volume of Hemisphere

Example 1: If the radius of a hemisphere is 21 cm. Find the volume of the hemisphere.

Solution:

We know that volume of hemisphere = 2/3 π r³

                                                          = 2/3 x 22/7 x 21 x 21 x 21

                                                          = 2 x 22 x 21 x 21

                                                          = 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:

volume of hemisphere = 2/3 π r³

Diameter = 2 x radius 

     radius = diameter / 2

Volume = 2/3 π (5/2)³

             = 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

                                    = 2/3 x (3.14) x (2)³

                                    = 16.75 cm³

volume of water = volume of cuboid – volume of hemisphere 

                          = 32 – 16.75

                          = 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.

FAQs on Volume of Hemisphere

Question 1: What is a Hemisphere?

Answer:

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.

Question 2: If a sphere and a hemisphere have the same radii then what is the ratio of their volume?

Answer:

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

Question 3: How do we measure the volume of the hemisphere?

Answer:

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.

Question 4: Write the formula for the surface area of the hemisphere.

Answer:

The surface area of any hemisphere is given by:

• CSA = 2πr²
• TSA = 3πr²

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