Volume of a Sphere — Formula, Derivation & Worked Examples

Many students always remain perplexed when it comes to the calculation of the volume of a sphere. In a simplified manner, the volume of a sphere is the accurate calculation of the total area it can capture. The figure of a sphere is of 3 dimensions and with no sides. Often, students consider the calculation quite tricky, but if you are aware of the proper techniques and formulas, you will never get confused in the future.

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Moreover, we promise to deliver 100% original content that excels among others. From research to referencing formatting structuring, we do everything in the best way. But, still, we advise you to understand this concept and try to solve questions related to it.

To make the process even smoother for you, we have come up with this write-up to provide you with a perfect understanding of the topic. After going through this streamlined information, you will definitely get a hold of this concept.

Conceptual Understanding Of The Volume Of Spheres

The first and foremost thing to know before understanding this concept is the relationship between the radius and volume of a sphere. Radius is directly proportional to the volume in the case of a sphere. A minor change in the radius can drastically change the volume. The unit that calculates the volume of a spherical body is (unit)^3. Metric units of the volume of a spherical object are cubic meters and cubic centimeters.

Kind of Spheres

There are two kinds of spheres:

1.     Solid sphere

2.     Hollow sphere

Different processes are followed to calculate the volume of two different kinds of spheres. In the following part, you will get a thorough understanding of this aspect. Once you get an idea about the topic, and especially if you are well aware of the derivation, it will be extremely easy for you to handle all the sums related.

Derivation Of The Volume Of A Sphere

As per the Archimedes principle, one can presume the volumes of a cone, cylinder, and sphere to be in the ratio of 1:2:3 if they have the same radius ‘r’ and also have the same cross-sectional area. With respect to the above theorem, it is quite obvious that the following linkage can be made between the volume of a sphere, the volume of a cylinder, and the volume of a cone:

The volume of the cylinder = Volume of the Cone + Volume of the Sphere.

Subsequently, you can derive that: Volume of Sphere = Volume of Cylinder- Volume of Cone.

As, the cylinder volume= πr2h and the cone volume= (1/3) πr2h,

The volume of the sphere =Volume of Cylinder - Volume of Cone

πr2h- (1/3) πr2h= (2/3) πr2h

If, the height of cylinder = diameter of sphere= 2r

Therefore, the formula of a sphere’s volume is (2/3) πr2h= (2/3) πr2(2r)= (4/3) πr3.

The Formula for Volume of a Sphere

The volumes of the two kinds of spheres are calculated differently. In the case of a solid sphere, there is only one radius, whereas when it comes to hollow ones, there are 2 radii. For a hollow sphere, there are two different values of radii (one is inner radius, and the other is outer radius). In order to ease your task, you need to apply the following formula. 

● The volume of a solid sphere

Suppose the sphere's radius is ‘r.’ Let's assume the volume of the sphere is ‘V.’  In this case, you can derive the equation for a sphere’s formula:

The volume of Sphere, V= (4/3) πr3.

● The volume of a hollow sphere

Consider the radius of the outer sphere to be “R,” the radius of the inner sphere is “r,” and the sphere’s volume is V, then the volume of the sphere will be:

Volume of Sphere, V= Volume of Outer Sphere - Volume of Inner Sphere= (4/3) πR3– (4/3) πr3 = (4/3) π (R3 – r3).

Steps to Calculate The Volume Of A Sphere

As mentioned above, the capacity within a sphere largely decides its volume. Let's have a look at how to calculate the volume of a sphere formula listed below:

Step 1: Look for the value of the radius.

Step 2: Take the radius cube.

Step 3: Take a product of r3 and (4/3) π

Step 4: Add all the units to get the final answer.

Let’s go through some examples to get a clear understanding:

1. Measure the volume of a sphere of 4 inches radius.

As mentioned above,, the volume of the sphere, V= (4/3) πr3 with r= 4 inches

V= ((4/3) × π × 43) in^3

V= 268.08 in3

2. Find the volume of the sphere with a diameter of 10 cm.

If diameter = 10 cm, the radius will be = (10/2) cm= 5 cm.

Now, put this value in the sphere volume formula:

V = 4/3 π 53

V = 4/3 x 22/7 x 5 x 5 x 5

V = 4/3 x 22/7 x 125

V = 523.8 cm3

Therefore, now that you have understood how to find the volume of a sphere, the calculations will be effortless for you. All that you need is to be acquainted with the formula. Proceed further with the blog and get to learn more about this complicated concept more easily.

Some Solved Examples to Grasp the Concept Even Better

Solved Examples

 

Solution: As, Mass = Volume x Density.

Since the shot-putt is a metallic solid sphere,

As per the formulae for the volume of a sphere:

Volume = 4/3 πr^3

             = 4/3π (4.9)^3 = 493 cm^3

Mass = Volume×Density

         = 493×7.8 g

         = 3845.44 g

         = 3.85 kg

Hence, the mass of the shot-putt comes out to be 3.85 kg.

Solution:

Volume =4πr^3

          => 4πr^3 = 5000

          => r = 5000×3/4π

          => r = 10.61 cm

 

Solution:

The radius of the hemisphere = 6cm

The volume of a hemisphere can be calculated as

Volume = ⅔  π x radius³ cubic units.

           = ⅔ x 3.14 x 6 x 6 x 6

             = 452.16 cubic cms.

As a result, finally, the volume of the hemisphere comes out as  452.16 cubic cm.

Solution:

Volume = 2500cm³

The volume of the hemisphere = ⅔ π x r³

                                                  ⇒ 2500 = ⅔  π x r³

                                                  ⇒ 2500 x 3 = 2πr³

                                                  ⇒ r³ =7500/2×π

                                                  ⇒ r³=7500/2×3.14

                                                  ⇒ r³ =7500/6.28

                                                  ⇒  r³ = 1194.267

                                                  ⇒ r³ =(1194.267)^½

                                                  ⇒ r = 10.6096358

              Hence, the radius of the hemisphere is 10.6096358.

 

1. Measure the Mass of a Shot-putt (metallic sphere) of Radius 4.9cm. The Density of the Metal is 7.8gcm³.
2. Determine the Radius of a Spherical Ball Whose Volume is 5000cm³.
3. Determine the Volume of the Hemisphere With a Radius of 6 cm.
4. The Volume of a Hemisphere is 2500 cm². Find the Radius of the Hemisphere.
5. 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= 4/3 × Π × R³

  = 4/3 × 22/7 ×21 ×21 ×21

  =V = 4 × 22 × 21 × 21

            ->V= 38808 m³

Since,

1 m³ = 1000 liter

Then, the volume of the tank,

= 38808 m³ × 1000

= 38808000 liters.

So, 38808000 liters of water is the capacity in the tank.

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