Did you observe why the cricket ball feels heavier than the tennis ball? Or why is a tiny marble difficult to break? The answer lies in their volumes. Though they are made of different materials, their volumes play a significant role in determining their weights. Volume is the space inside a three-dimensional object or shape. Two-dimensional objects will not have volume. The volume of a sphere means the capacity it can hold. What does it mean, or what is its formula? Let’s find out!

The equation for the volume of a sphere

As mentioned above, the volume of a sphere is the capacity it can hold. If a sphere is cut into two parts, the inside space or the filler material inside it will be its volume. If there is a space inside, then the sphere is hollow. Whereas if the filler material is inside, the sphere will be solid.

The volume of a sphere is determined by the three coordinates x, y, and z. Why? Because a three-dimensional object will lie on all three axes. Volume is measured in cubic meters, cubic feet, cubic inches, and similar units. It is represented by symbols cm3,m3,in3, etc.

Then, how do you find the volume of a sphere? Well, it depends on the diameter of the sphere. If the surface area is multiplied by the diameter, the volume will be obtained, in which every point on its surface is equidistant from its center. Mathematically, to calculate the volume of a sphere, the following formula is used:
The volume of a sphere = 4/3 𝜋 r³, where r is the radius of the sphere.

Volume is a fixed quantity and can be found using Archimedes’ principle. According to Archimedes, if the solid sphere is dropped in a container filled with water, the volume of water displaced will be equal to the volume of the sphere. The volume will change when the values of the diameter or radius of the sphere will change. Otherwise, the formula for the volume of the sphere will remain the same.

Is this method always relevant to measuring the volume? Do we have to keep a tub of water every time to measure the volume? Now, coming to how it is derived? Where did this volume formula come from? For this, it’s time to go the elementary method and learn some other 3D shapes.

Where does the volume of a sphere come from?

The equation for the volume of a sphere can be derived from the integration method and the volumes of the cone and cylinder.

Method 1: Integration method
Consider a sphere with numerous thin spherical discs arranged over another as depicted in the pictorial representation. Since the sphere is made with thin circular discs placed collinearly, their diameters will vary over the entire length of the sphere. As a result, the volume will also change over the whole diameter of the sphere.
Integration method
Now, consider a thin disc with radius r and thickness dy, located at a distance of y from the x-axis. Also, the volume of the sphere will be the product of the area of the circle and its thickness. We can represent the radius of the circular disc r in terms of y using the Pythagorean theorem.
Therefore, the volume of the disc element, dV, can be given by:
dV = r² dy
dV = 𝜋 (R² – y²) dy

By integrating the above equation, the total volume of the sphere will be given by:

Therefore, the final formula for the volume of a sphere is given by V = 4/3 𝜋 r³.

Method 2: From volumes of cone and cylinder
Do you know the sphere, cylinder, and cone have a connection? Precisely, their volumes have a connection! The volume of a cylinder is the sum of the volume of a cone and the volume of a sphere. Mathematically,

Vcylinder = Vcone + Vsphere

Therefore, we can find the volume of a sphere as Vsphere = Vcylinder – Vcone
Vcone = 𝜋r2h, where h is the slant height of the cone.
Vcylinder = ⅓ 𝜋r2h, where h is the height of the cylinder.
Therefore, Vsphere = ⅓ 𝜋 r2h – 𝜋r2h = 2/3 𝜋r2h

If we observe a sphere, we can see that the height is equal to the sphere’s diameter. Therefore, h = 2r.
Putting the value of h in the final equation, we get,
Vsphere = ⅔ 𝜋r2 (2r) = 4/3 𝜋 r3, which is the volume of a sphere.

Learning about the volume of a hollow sphere

Moreover, the volume of a hollow sphere is related to the volume of a sphere. In a hollow sphere, the outer radius is represented by R and the inner radius by r. Then, the volume of a hollow sphere is given by,
Vhollow = 4/3 𝜋 R3 – 4/3 𝜋 r3
It can also be written as Vhollow = 4/3 𝜋 (R3 – r3). The unit of volume of the hollow sphere is cubic meters.

How to find the volume of a sphere?

After learning about the volume of a sphere, all we have to do is find the volume! Anyone can find the volume of a sphere without using the volume of a sphere calculator. Follow these steps and find the volume:

Step 1: Go through the data provided in the question carefully.
Step 2: Check which value is given; radius, diameter, surface area, or circumference.
Step 3: Find the radius of the sphere. If the diameter is given, divide it by 2 to find the radius. If the surface area is given, find the value of radius from the surface area of a sphere formula 4𝜋r. If the circumference is given, find the radius from the formula 2𝜋r.
Step 4: Go through the units carefully. Convert all the units equivalent to each other in one single form.
Step 5: Get the cube of the radius, i.e., r³.
Step 6: Multiply the value of r³ by 𝜋.
Step 7: Multiply the value found in Step 6 by 4/3.
Step 8: The final value will be the required volume of a sphere.

Applications of the volume of a sphere in the real world

In the real world, the volume of a sphere is used in several ways. If we know its formula, we don’t need a volume of a sphere calculator to calculate it every time. Here are a few applications where the volume formula is used frequently:

  1. The volume formula is used in many industries while manufacturing objects, like balls, globes, bearings, bubbles, etc.
  2. It is helpful to calculate the amount of air required to preserve leakage in a hot air balloon.
  3. Calculating volume is necessary if carrying any harmful chemical in a spherical container.
  4. The volume of a hollow sphere is used to know the quantity of any material kept in a bowl or semi-spherical shell.

Time to learn the volume of a sphere with examples

Example 1: Find the volume of a sphere whose circumference is 144 units.
Solution: Given that the circumference of a sphere is 144 units.
We know the circumference of a circle is given by 2𝜋r, where r is the radius.
Therefore, C = 2𝜋r = 144
Solving this, we get r = 22.9 units.
The formula gives the volume of a sphere, V = 4/3 𝜋 r3
Putting the value of r, we get V = 4/3 𝜋 (22.92)3
V = 50453.197 unit³.

Example 2: A hollow sphere is designed by a company such that its thickness is 10 cm and 6 m inside diameter. What will be the volume of the sphere designed by the company?
Solution: Given that the inside diameter is 6 m and thickness is 10 cm, equal to 0.1 m.
Therefore, the outer diameter will be 6 + 0.1 m = 6.1 m.
The volume of a hollow sphere is denoted by: Volume = 4/3 𝜋R3 – 4/3 𝜋r3, where R is the radius of the outer sphere and r is the radius of the inside sphere.
Putting the values in the above equation, we get,
V = 4/3 𝜋 (3.053 – 33) = 4/3 𝜋 (1.37)
Therefore, V = 5.735 m3.

Example 3: Find the volume of a sphere if its surface area is 100 square meters.
Solution: We know that the surface area of a sphere is given by S = 4𝜋r, where r is the radius of a sphere.
Therefore, S = 4𝜋r = 100
Finding the value of r, we get, r = 7.96 m
The volume of a sphere is given by V = 4/3  𝜋 r3
Putting the value of r, we get,
V = 4/3 𝜋 (7.96)3 = 2111.58 m3.