Let’s explore the Volume of a Sphere!

In many cases, we need to use the volume of a sphere in various areas, calculations, etc. So, what is the meaning of it?

In this article, we will learn the details of the volume of a sphere, its definition, formula or equation calculation, unit in Gallons or liters, along with so many examples.

Let’s explore!

I am sure that you have seen football or even cricket ball. So, how do these football or cricket balls like?

This is simple! This object looks like a sphere!

Many other objects for the sphere can be as follows,

- Basketball
- Planet earth
- Globe, etc.

Now, we got a brief idea about the sphere, so what about the volume of a sphere object-like football or globe?

It is simple, the volume of the sphere means the amount of space or materials the sphere acquires. Let’s try to understand sphere volume with a simple example.

Let’s take a football which is a spherical object to understand the meaning of spherical volume. Now, take a bucket with a volume mark and put some water into it. Say, the bucket has water of around 10 liters water.

Now, the next step is to keep the football within the water fully. You will observe that the water level in the bucket is increased to 16 liters.

We all know from Archimedes’ principle, that the sphere has replaced the same volume of water. This increased volume of water is nothing but the volume of the football as well as we can say that, its volume of a sphere.

- This is not a direct method which we normally adopt in sphere volume calculation. Then what do we do to calculate its volume then?
- We calculate the volume by a simple equation or a formula. We will explore it
- It is not to be necessary to have liquid, and you have to put a sphere there to calculate the volume. With the help of a simple equation or formula, we can easily calculate the volume within a few seconds.

The volume of a sphere can be defined as the capacity of an object of round shaped as well as 3-dimensional (3D) and the distance from the center to each point on the surface is the same.

It can also be defined as a geometrical shape with round shapes as well as 3-dimensional (3D).

A sphere has three axes,

- X – axis
- Y – axis
- Z – axis

Look at this below image of spheres, here,

- Round shaped
- 3D objects

We have understood the primary concept of the volume of a sphere. Now, if you want to calculate the volume fast, then there should be a formula or equation of sphere volume, right!

- This equation or formula is used to calculate the capacity of the sphere.
- The amount of material can get from a sphere.
- It indicates the space availability or the amount required for a sphere.
- The surface area of the sphere can also be calculated.
- Radius can also be calculated from a given volume of a sphere.

Formula or equation for the volume of a sphere can be written as,

- Volume of a sphere, V = 4/3 x π × (Radis)
^{3} - V = 4/3π x r
^{3} - V = 4/3πr
^{3}

Where,

- r = Radius of the sphere

So, it is super easy, right? Get a radius and put the same in the sphere formula! But, do you want to know how to derive the formula of the sphere? Let’s get in!

The derivation of the sphere formula is simple, but you need to have a slight idea about integration! It can be proved without integration as well, however, with integration the proof is easy!

Let’s take, a sphere-like football. We will derive the formula of this football or say the sphere. Refer to our below diagram.

The radius of the sphere = r

Consider the center of the sphere at the origin and cut a section or a slice from y distance.

Here,

- y = the distance between x-axis to section
- x = radius of the slice (as it is a round shape)
- r = radius of a sphere or the hypotenuse of the triangle OAB

Now, can we find out the area of the slice?

Yes, we can!

Just we need to know the radius, x.

So, how to find out the radius? Let us look!

Based on Pythagorean theorem, we all know that,

- x
^{2}+ y^{2}= r^{2} - x
^{2}= r^{2 }– y^{2} ^{x =}√ (r^{2 }– y^{2})

Hence, we got the value of x in terms of r and y.

As we got the value of x and can easily find out the area of that slice.

- Area A = π x (radius)
^{2} - Area A = π x (x)
^{2}[as the radius of the slice is x] - Area A = π [√ (r
^{2 }– y^{2})]^{2} - Area A = π (r
^{2 }– y^{2})

The sphere is consisting of so many slices vertically. We have got only one area of slice so how do we get all other slices to conclude the volume of the entire sphere?

It can be done by definite integral!

Let’s consider, dy is a very small height along vertical direction and the range is from ‘-r’ to ‘+r’.

So, the volume of that slice with dy height, dV

- dV = Area x Height
- dV = A . dy
- dV = π (r
^{2 }– y^{2}) . dy - dV = π (r
^{2 }– y^{2}) dy

By integrating,

- V = ∫ [π (r
^{2 }– y^{2}) dy] ranges ‘-r’ to ‘+r’ - V = π ∫ (r
^{2 }– y^{2}) dy ranges ‘-r’ to ‘+r’ - V = π[∫r
^{2}dy – ∫z^{2}dy] ranges ‘-r’ to ‘+r’ - V = π [y.r
^{2 }– y^{3}/3] ranges ‘-r’ to ‘+r’ - V = π [{(r.r
^{2 }– (r)^{3}/3} – {(-r.r^{2 }– (-r)^{3}/3}] - V = π [{(r
^{3 }– r^{3}/3 + r^{3 }– r^{3}/3}] - V = π [{(2r
^{3 }– 2r^{3}/3}] - V = π [4r
^{3}/3] - V = 4/3πr
^{3}

Hence, this is the volume of a sphere formula or equation.

Now, let’s see what is the formula of a sphere with the help of diameter. We all the relationship between radius and diameter.

- Diameter, D = 2 x Radius
- Diameter, D = 2 x r
- Diameter, D = 2r
- Or 2r =D
- Or r = D/2

Volume of sphere, V_{D}

- V
_{D}= 4/3πr^{3} - V
_{D}= 4/3π(D/2)^{3}[as r=D/2] - V
_{D}= 4/3π(D^{3/}8) - V
_{D}= 1/6πD^{3}

Hence, volume of sphere with radius, V_{D }= 1/6πD^{3}

We have already derived the equation of the volume of a sphere. Now, what about the volume of a hollow sphere?

Can we use the same formula over here?

No, we cannot!

Let’s see what is the volume of a hollow sphere!

A hollow sphere means it has basically two radiuses.

- The first one is the radius for the outer sphere, say, R
- The second one is the radius for the inner sphere, say, r

Now, the volume of the hollow spherical object implies the difference between the volume of the outer sphere and the inner sphere.

- Volume of outer sphere = V
_{R}= 4/3πR^{3} - Volume of inner sphere = Vr = 4/3πr
^{3}

So, the difference between outer and inner sphere volume, V_{h}

- V
_{h}= V_{R}– V_{r} - V
_{h}= 4/3πR^{3}– 4/3πr^{3} - V
_{h}= 4/3π (R^{3}– r^{3})

The unit of volume of a spherical object can be, as follows

The volume of a Spherical object = 4/3πr^{3}

The unit of volume of a spherical object = 4/3π x (unit of radius)^{3 }

In S.I unit

- The S.I unit of volume of a spherical object =4/3 π x (m)
^{3 }= m^{3 }[π is unit less]

In C.G.S unit

- The C.G.S unit of volume of a spherical object = 4/3 π x (cm)
^{3 }= cm^{3 }or cc [π is unit less number] - 1 litre = 1000 cm
^{3 }= 1000 cc

In F.P.S unit

- The F.P.S unit of volume of a spherical object = 4/3 π x (ft)
^{3 }= ft^{3 }[π is unit less number] - number]

Let’s see the units of volume of the sphere in U.S gallons (widely used).

- 1 liter = 0.264 U.S. gallons [Remember this value]
- or, 1000 cm³ = 0.264 U.S. gallons [ as 1 liter = 1000 cm³]
- 1000 000 cm³ = 264 U.S. gallons [ Multiplying 1000 in both side]
- 1 m³ = 264 U.S. gallons [ as 1 m³ =1000 000 cm³]
- 3.28 x 3.28 x 3.28 ft³ = 264 U.S. gallons [ as 1 m = 3.28 ft]
- 1 ft³ = 264/(3.28 x 3.28 x 3.28) = 7.48 U.S. gallons
- 1 milliliter = 1/1000 liter = 264/1000 U.S. gallons = 0.264 U.S. gallons

Based on the formula, we have understood that if we get the value of the radius of the sphere, we can easily calculate the sphere volume.

So, the volume of a sphere can be calculated based on a few simple steps,

- Find out the Radius of the sphere
- Calculate the volume formula or equation with units

**Step#1 Radius of Sphere**

Firstly, as in the sphere volume, there is only one variable, which is the radius. So, we must get the value of radius.

- If you have the given data, then use it. Say this value is r.
- If the diameter is given, then radius can be calculated from, D = 2r formula. So, r =D/2.
- In case, the surface area, A’ is given, we can find out the radius from it. We know, A’=4πr
^{2}, or r = 1/2√(A’/π).

Let’s see a few examples,

- a sphere has a diameter, D = 40cm. So, the radius of sphere, r = D/2 = 40/2 = 20 cm.
- a sphere has a surface area, A’ =400π cm. So, it’s radius, r = 1/2√(A’/π) = 1/2√(400 π /π) = 10 cm.

**Step#2 Calculation of Sphere Volume**

As we have got the radius, we can easily calculate the volume from the formula.

Volume, V = 4/3πr^{3}

Now, we can calculate and check the volume by putting the radius into the equation. We have got two radius in our examples,

Radius = 20 cm

Radius = 10 cm

Let’s see the volume for both,

- The volume of spherical object of having 20 cm radius = 4/3πr
^{3 }= 4/3π 20^{3 }= 33493 cm^{3}= 33493 cc = 33.493 liters [π = 3.14] - The volume of spherical object of having 10 cm radius = 4/3πr
^{3 }= 4/3π 10^{3 }= 4187 cm^{3}= 4187 cc = 4.187 liters [π = 3.14]

There are so many examples of spherical objects in our day-to-day life. Just remember or derive the formula and calculate the volume for applications.

Various manufacturers have used this formula to design different types of objects.

Let’s see a few examples of sphere volumes,

- Spherical globe
- Moon
- Football
- Marbles
- Tennis ball
- Planets

Let’s see a couple of problems for the volume of sphere calculations and understand the concept clearly with examples.

**Question 1**

Find out the volume in liters of a sphere with a radius of 13 cm.

**Solution**

Input data

- Sphere radius of football, r = 13 cm

As per the formula of volume of football, V = 4/3πr^{3 }

Hence, volume, V

- V = 4/3πr
^{3} - V = 4/3 x 22/7 × 13
^{3} - V = 9198 cm
^{3}

As the radius is in cm, so, the volume will be coming in cm^{3}, and we need the volume in liters.

We already know that 1 liter = 1000 cm^{3}

Or, we can write, 1000 cm^{3} = 1 liter

∴ 9198 cm^{3} = 9198 / 1000 = 9.198 liters

The volume of the football is 9.198 liters.

**Question 2**

Find out the volume in a cubic meters of a sphere with a diameter of 180 cm.

**Solution**

Input data

- The diameter of the sphere, D = 180 cm

From the input data, we can find out the radius, r

- D = 2r
- r = D/2
- r = 180/2
- r = 90 cm

As per the formula of volume of spherical object, V = 4/3πr^{3 }

Hence, volume, V

- V = 4/3πr
^{3} - V = 4/3 x 22/7 × 90
^{3} - V = 305208 cm
^{3}

As the radius is in cm, so, volume needs to be in m^{3 }as per question.

So, we already know that 1 m^{3 }= 1 m x 1 m x 1 m = 100 cm x 100 cm x100 cm = 1000000 cm^{3}

Or, we can write, 1000000 cm^{3} = 1 m^{3}

∴ 3052080 cm^{3} = 3052080 / 1000000 = 3.05 m^{3}

The volume of the spherical object is 3.05 m^{3}.

**Question 3**

Find out the radius of a spherical object of volume 4000 cm^{3}.

**Solution**

Input data

- Sphere volume, V = 4000 cm
^{3}

As per the formula of volume of spherical object, V = 4/3πr^{3 }

Hence, volume, V

- V = 4/3πr
^{3} - 4000 = 4/3 x 22/7 × r
^{3} - 4/3 x 22/7 × r
^{3 }= 4000 - r
^{3 }= 4000 / (4/3 x 22/7) - r
^{3 }= 4000 x ¾ x 7/22 - r
^{3 }= 955 - r = 9.85

Hence, the radius of the sphere is 9.85 cm.

**Question 4**

Find out the volume of a sphere of surface area 200 cm^{2}.

**Solution**

Input data

- Surface area of sphere, A’ = 200 cm
^{2}

As per the formula of surface area, A’ = 4πr^{2 }

Hence, surface area, A’

- A’ = 4πr
^{2} - 200 = 4πr
^{2} - 4πr
^{2 }= 200 - r
^{2 }= 200/4π - r = √(200/4π)
- r = 4 cm

So, volume shall be as follows,

Volume, V

- V = 4/3πr
^{3} - V = 4/3 x 22/7 × 4
^{3} - V = 268 cm
^{3}

Hence, the volume of the spherical object is 268 cm^{3}.

**Question 5**

Find out the volume of a hollow spherical object with an inner and outer radius of 10 cm and 20 cm respectively.

**Solution**

Input data

- The outer radius of the sphere, R = 20 cm
- The inner radius of the sphere, R = 10 cm

As per the formula of volume of spherical object, V_{h} = 4/3π (R^{3} – r^{3})

Hence, volume, V_{h}

- V
_{h}= 4/3π (R^{3}– r^{3}) - V
_{h}= 4/3π (20^{3}– 10^{3}) - V
_{h}= 4/3π [ (2.10)^{3}– 10^{3}] - V
_{h}= 4/3π 10^{3}[ (2)^{3}– 1^{3}] - V
_{h}= 4/3π 10^{3}[ 7] - V
_{h}= 4/3 x 22/7 x 10^{3 }x 7 - V
_{h}= 4/3 x 22 x 10^{3 } - V
_{h}= 29.333 x 10^{3 } - V
_{h}= 29333

The volume of the hollow spherical object is 29333 cm^{3}.

Hence, we have got the basics of the volume of a sphere shape, calculation steps, along with many examples.

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