Learn About the Volume of a Sphere
A circle can be drawn on 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, for example- Ball, Earth, etc. A Sphere is a 3D figure whose points lie in 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:
Solid Sphere - A solid object in the form of a sphere is called a solid sphere. It is more like a sphere filled up with the same material it is made up of.
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 a 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 center). 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
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Derivation of the Formula of the Sphere
Archimedes was very fond of spheres and cylinders. 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 the 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 = πr2h
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 the trough to the 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.
Solved Examples
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.
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
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.
The Volume of a Hemisphere is 2500 cm2. Find the Radius of the Hemisphere.
Solution:
Volume = 2500cm3
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.
A spherical shaped tank has a radius of 21 m. Now find the capacity of it in a liter to store water in it.
Solution:
Given values are,
R = 21 m
Now the volume of a sphere,
V= \[\frac{4}{3}\] × Π × R3
V = \[\frac{4}{3}\] × \[\frac{22}{7}\] × 21 × 21 × 21
V = 4 × 22 × 21 × 21
V= 38808 m3
Since,
1 m3 = 1000 liter
Then the capacity of the tank,
= 38808 m3 × 1000
= 38808000 liter.
So, 38808000 Liter water can be stored in the tank.
The article is useful for the students to grab the concept of geometry. It provides the formula of volume of the sphere along with the derivation and solved examples based on the sphere.
FAQs on The Volume of a Sphere
1. What is half of the Sphere called?
Half of the Sphere is called the 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. We know that the volume of the sphere is = 1/2(4/3 πr3 ) = 2/3 πr3
where r is the radius.
2. How to Calculate Volume of Sphere?
The volume of a sphere is basically the space occupied inside a sphere. The volume of the sphere can be defined using the formula of the volume of the sphere. The steps for calculating the volume of a sphere are:
First check the value of the radius of the sphere.
Then take the cube of the radius.
Now multiply r3 by (4/3)π
Then, at last, add the units to the final answer.
3. What is the Difference between two Dimensional and Three Dimensional?
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).
4. State some real-life examples of Volume of a Sphere?
For drawing a circle on a sheet of paper, take a circular disc, paste a string along its diameter and rotate it along the string and this will give the shape of a sphere. In day to day life, there are different types of spheres like Basketball, football, table tennis, etc. these all are some of the common sports that are played by people all over the world. The balls that are used in these sports are nothing but spheres of different radius. The volume of sphere formula is also useful in designing and calculating the capacity or volume of such spherical objects.
5. What is the Area and Volume of a Sphere?
The surface area of a sphere is defined as the total area or region covered by the surface of the sphere. The surface area of a sphere is given by:
The surface area of the sphere = 2πrh.
And if the diameter of the sphere = 2r, then the surface area of the sphere = 4πr2 square units.
The area of a sphere is only expressed in square units, like m2, in2, cm2, yd2, etc. The volume of a sphere is defined as the total amount of capacity applied in a sphere that can be calculated using the volume formula for the sphere which is V = (4/3)πr3. The volume of a sphere is measured in cubic units only.
