The Volume of a Sphere

A circle can be drawn on a paper but a sphere can't be drawn on a piece of paper. This is because Circle is a two-dimensional figure whereas a sphere is a three-dimensional object, example- Ball, Earth, etc. A Sphere is a 3D figure whose all the points lie in the space. All the points on the surface of a sphere are equidistant from its centre. This distance from the surface to the centre is called the radius of the sphere. 

 

Types of Spheres

Spheres are of Two Types:

  1. Solid Sphere - A solid object in the form of the sphere is called a solid sphere. It is more like a sphere filled up with the same material it is made up of.

  2. Hollow Sphere - If a solid sphere is cut and taken out of a big solid sphere, leaving behind a thin surface in the form of a spherical shell is called Hollow sphere. It is more like a balloon or ball filled with air.

 

What is the Volume of a Sphere?

The volume of a sphere is the three-dimensional space occupied by a sphere. This volume depends on the radius of the sphere (i.e, the distance of any point on the surface of the sphere from its centre). If we take the cross-section of the sphere then the radius can be calculated by reducing the length of the diameter to its half. Or we can also say that the radius is half of the diameter. 

What is the Formula for the Volume of a Sphere?

Suppose if the radius of a sphere is ‘r’ then the volume of sphere formula is, 

The Volume of a Sphere = 4/3 πr³

Derivation of the Formula of the Sphere

Archimedes was very fond of sphere and cylinder. He did one of the most remarkable mathematical deductions and used the earlier concept of Egyptian and Babylonian to find the volume of a sphere. The derivative of volume of a sphere found its origin from the subdivision of the volume of cone, sphere and cylinder of the same cross-sectional area into slices and came to a conclusion that the sum of the volume of a cone and sphere is equal to the volume of a cylinder of same cross-sectional area. The volumes are in the ratio 1:2:3 respectively.

Also, the volume of two cones of radius ‘r’ is equal to the volume of the sphere of the same radius. That is, it takes two cones of water to fill up a sphere of the same radius as that of a cone. Let’s find the derivative of the volume of a sphere.

We know, 

The volume of a cylinder = \[\pi {r^2}h\]

The volume of cylinder = Volume of cone + Volume of Sphere

Replacing the volume of the sphere,  (Volume of sphere = Volume of cone + Volume of the cone)

The volume of cylinder = 3\[ \times \] volume of the cone

\[\pi {r^2}h = 3 \times \]Volume of cone

\[\frac{{\pi {r^2}h}}{3} = volume{\text{ }}of{\text{ }}cone\]

Now. according to the volume of a sphere proof

The volume of a sphere = Volume of a cone + Volume of a cone.

That is, the volume of a sphere = \[ = \frac{{\pi {r^2}h}}{3} + \frac{{\pi {r^2}h}}{3}\]

The height of the cone = diameter of sphere = 2r

Thus, replacing h = 2r

The volume of the sphere  \[ = \frac{{2\pi {r^3}2r}}{3} + \frac{{2\pi {r^3}}}{3}\]

The volume of the sphere \[ = 2 \times \left( {\frac{{2\pi {r^3}}}{3}} \right)\]

The volume of the sphere \[ = \frac{{4\pi {r^3}}}{3}\]

The volume of the sphere =\[\frac{{4\pi {{\text{r}}^3}}}{3}\]

Therefore the Equation for the Volume of a Sphere is \[ = \frac{{4\pi {r^3}}}{3}\].

The Volume of a Sphere of Unknown Radius

Take a sphere of unknown radius, a container, a trough and a measuring cylinder. 

Arrange the apparatus as shown below. Fill the container with water up to the brim and then carefully place the sphere in the container. This will make some water flow out of the container which will get collected in the trough.

Pour the water from trough to measuring cylinder to find out the amount of water displaced by the sphere.

The amount of displaced water is equal to the space occupied by the sphere.

The volume of sphere = Amount of displaced water.

Examples: 

1. Calculate the Mass of a Shot-putt (metallic sphere) of Radius 4.9cm. The Density of the Metal is 7.8gcm3.

Solution: We know, Mass = Volume x Density.

Since, the shot-putt is a metallic solid sphere,

According to equation for volume of a sphere:

Volume = \[\frac{{4\pi {r^3}}}{3} = \frac{{4\pi {{\left( {4.9} \right)}^3}}}{3} = 493c{m^3}\]

Mass = Volume \[ \times \] Density

          = 493 \[ \times \] 7.8 g

          = 3845.44g

          = 3.85 kg

Therefore, the mass of the shot-putt is 3.85 kg. 

 

2. Find the Radius of a Spherical Ball Whose Volume is 5000cm3.

Solution: 

Volume = \[\frac{4\pi r^{3}}{3}\]

\[\frac{4\pi r^{3}}{3}\] = 5000

\[r^{3}\] = \[\frac{5000\times 3}{4\pi}\]

r = 10.61 cm

 

3. Determine the Volume of the Hemisphere Having a Radius of 6 cm?

Solution:

The radius of the hemisphere = 6cm

The volume of a hemisphere can be expressed as:

Volume = \[\frac{2}{3}\] π x radius³ cubic units.

               =  \[\frac{2}{3}\] x 3.14 x 6³

                =  \[\frac{2}{3}\] x 3.14 x 6 x 6 x 6

= 452.16 cubic cms.

Therefore, the volume of the hemisphere is 452.16 cubic cms.

 

4. The Volume of a Hemisphere is 2500\[c{m^2}\]. Find the Radius of the Hemisphere.

Solution:

Volume = 2500

The volume of hemisphere = \[\frac{2}{3}\] π x radius³

⇒ 2500 = \[\frac{2}{3}\] π x r³

⇒ 2500 x 3 = 2πr³

⇒ r³ = \[\frac{7500}{2 \times π}\]

⇒ r³ = \[\frac{7500}{2 \times 3.14}\]

⇒ r³ = \[\frac{7500}{6.28}\]

⇒ r³ = 1194.267

⇒ r³ = \[\sqrt[3]{1194.267}\]

⇒ r = 10.6096358

Therefore, the radius of the hemisphere is 10.6096358.

FAQ (Frequently Asked Questions)

Q1) What is half of the Sphere called?

Answer: Half of the Sphere is called Hemisphere. The prefix ‘Hemi’ means half. A sphere is defined as a set of three-dimensional points and the centre is equidistant with all the points on the surface. Just for say, when an aircraft takes a round of the earth from earth’s midpoint, it covers two hemispheres of the earth. It can be said that the globe is made of two hemispheres. A globe generally produces two hemispheres exactly. Our earth is composed of two hemispheres, the southern and the northern hemispheres.

Q2) How to find the Volume of a Hemisphere?

Answer: Hemisphere is half of the sphere. We know that the volume of the sphere is

= 1/2[4/3 πr3]

= 2/3 πr3

where r is the radius. 

Q3) What is the Difference Between two Dimensional and Three Dimensional?

Answer: A Two-dimensional shape has only length and breadth but no depth (can be represented by x and y-axis) whereas a three-dimensional shape has length, breadth and depth (can be represented by x, y and z-axis).