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A sphere is 3D or a solid shape having a completely round structure. If you rotate a circular disc along any of its diameters, the structure thus obtained can be seen as a sphere. You can also define it as a set of points which are located at a fixed distance from a fixed point in a three-dimensional space. This fixed point is known as the center of the sphere. And the fixed distance is called its radius.

In this section, we will obtain the formula to compute the volume of a sphere. Volume, as you know, is defined as the capacity of a 3D object. Volume of a sphere is nothing but the space occupied by it. It can be given as:

Volume of a spherevolume \; of \; a \; sphere = \frac{4}{3}\pi ^{3}

Where ‘r’ represents the radius of the sphere.

Derivation:

The volume of a sphere can alternatively be viewed as the number of cubic units which is required to fill up the sphere.

  1. Let us take up an activity to find out the volume of a sphere.
  • Volume of a sphereTake cylindrical container. Pour water into it until it is filled to the brim.
  • Place this container in a large trough.
  • Now dip a spherical ball with a radius of ‘r’ units into the cylinder.
  • You will observe that some quantity of the water is displaced from the cylinder and falls out into the trough.
  • Pour this displaced water trough in another cylinder with radius ‘r’ units and height ‘2r’ units.
  • We know that the volume of the water displaced by the ball must be equal to the volume of the spherical ball.
  • Now take a note of the amount of water in the second cylinder. You will observe that the volume occupied by water is two-third of the volume of the second cylinder.

Hence, volume of water filled in the second cylinder = \frac{2}{3}=\pi r^{2}\times 2r

Thus,

                                   Volume of a sphere of radius r =  \frac{4}{3}=\pi r^{3}

Alternatively, the formula for the volume of a sphere can also be derived as follows.

Consider a sphere of radius r and divide it into pyramids. In this way, we see that the volume of the sphere is the same as the volume of all the pyramids of height, r and total base area equal to the surface area of the sphere as shown in the figure.

Volume of a sphereThe total volume is calculated by the summation of the pyramids’ volumes.

Volume of the sphere = Sum of volumes of all pyramids

Volume of the sphere= \frac{1}{3}A_{1}r+\frac{1}{3}A_{2}r+\frac{1}{3}A_{3}r.....\frac{1}{3}A_{n}r

=\frac{1}{3}r(Surface\;area\;of\;a\;sphere)

=\frac{1}{3}\times 4\pi r^{2}\times r

Volume of the sphere = \frac{4}{3}\pi r^{3}

Volume of a Sphere Formula in Real Life:

In our daily life, we come across different types of spheres. Basketball, football, table tennis, etc. are some of the common sports that are played by people all over the world. The balls used in these sports are nothing but spheres of different radii. The volume of sphere formula is useful in designing and calculating the capacity or volume of such spherical objects. You can easily find out the volume of a sphere if you know its radius.

Solved Examples:

Question 1: A sphere has a radius of 11 feet. Find its volume.

Solution: Given,

r = 11 feet

We know that, volume of a sphere = \frac{4}{3}\pi r^{3}

Volume of the sphere= \frac{4}{3}\times 3.14\times 11^{3} = 5572.45 cubic feet

Question 2: The volume of a spherical ball is 343\;cm^{3}. Find the radius of the ball.

Solution: Given, volume of the sphere= 343\;cm^{3}

We know that, volume of a sphere= \frac{4}{3}\pi r^{3}

343\;cm^{3}

= \frac{4}{3}\pi r^{3}

\Rightarrow r^{3}=\frac{343\times 3}{4\pi }=\frac{343\times 3}{4\times 3.14 }=81.92cm^{3}

\Rightarrow r=4.34cm

The radius of the ball is 4.34 cm.

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