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Surface Areas and Volumes

    Volume of a Sphere

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9th class > Surface Areas and Volumes > Volume of a Sphere

Volume of a Sphere

Let's try an activity regarding the volume of a sphere.

Activity: Take spheres of different radii and a container filled with water upto the brim. Place the filled up container in a trough to collect the displaced water.

Now, one at a time, place the sphere in the container and let the displaced water fall into the trough. After every trial for different spheres, measure the amount of water overflown into the trough. Also, calculate the value for 43πr3 where r is the radius of the sphere used for that particular trial. What do you observe?

The amount of water displaced will be equal to the value of 43πr3. We also know that the amount of water displace is equal to the volume of the object. Therefore,

Volume of a Sphere = \frac{4}{3}ππ r3

where r is the radius of the sphere.

What about the volume of a hemisphere? We know that the hemisphere is one half of a split sphere. Thus,

Volume of a Sphere = \frac{2}{3}ππ r3

where r is the radius of the hemisphere.

Let's Solve

  1. A hemispherical bowl has a radius of 3.5 cm. The volume of water it contains is equal to ____ cm3

Note: Round off to the nearest whole number

Finding volume of water

  • Volume of bowl : where r is the radius.
  • Substituting values in the volume formula
  • We get the volume = cm3(Round off to the nearest whole number).
  • We have found the desired answer.

  1. The volume of a sphere of radius 11.2 cm is ____ cm3

Note: Round off to the nearest whole number

Finding volume of sphere

  • Volume of sphere : where r is the radius.
  • Substituting values in the volume formula
  • We get the volume = cm3(Round off to the nearest whole number).
  • We have found the desired answer.

  1. A shot-putt is a metallic sphere of radius 4.9 cm. If the density of the metal is 7.8 g per cm3, find the mass of the shot-putt.

Finding mass of shot-putt

  • Volume of shot-putt : where r is the radius.
  • Substituting values in the volume formula
  • We get the volume = cm3
  • Mass = × Volume. We can use this to find the mass.
  • Mass of shot-putt = kg. (Round off to two decimal places)
  • We have found the desired answer.

  1. A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank.

Finding volume of metal used

  • Volume of hemisphere : where r is the radius.
  • But since, we need to find the volume of the metal used, the volume becomes where a and b are the outer and inner radii, respectively.
  • The outer radius a = m
  • Substituting values in the formula
  • We get the volume of metal = m3(Round off to four decimal places)
  • We have found the desired answer.

  1. Twenty seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S1. Find:

(i) radius r1 of the new sphere

(ii) ratio of S and S1

Finding the above answers

  • Volume of sphere : where r is the radius.
  • Thus, volume of twenty seven solid sphere =
  • Let the new sphere radius = r1
  • By solving for the new radius
  • We get r1 =
  • Now, finding the required ratio: SS1
  • The ratio SS1 =
  • We have found the desired answer.