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Volume of Sphere

The volume of a sphere is the measurement of the space it can occupy. A sphere is a three-dimensional shape that has no edges or vertices. In this short lesson, we will learn to find the volume of a sphere, deduce the formula of volume of a sphere and learn to apply the formulas as well. Once you understand this chapter you will learn to solve problems on the volume of the sphere.

1.	What is the Volume of Sphere?
2.	Derivation of Volume of Sphere
3.	Formula of Volume of Sphere
4.	How to Calculate the Volume of Sphere?
5.	FAQs on Volume of Sphere

What is the Volume of Sphere?

The volume of sphere is the measure of space that can be occupied by a sphere. If we draw a circle on a sheet of paper, take a circular disc, paste a string along its diameter and rotate it along the string. This gives us the shape of a sphere.

The unit of volume of a sphere is given as the (unit)³. The metric units of volume are cubic meters or cubic centimeters while the USCS units of volume are, cubic inches or cubic feet. The volume of sphere depends on the radius of the sphere, hence changing it changes the volume of the sphere. There are two types of spheres, solid sphere, and hollow sphere. The volume of both types of spheres is different. We will learn in the following sections about their volumes.

Derivation of Volume of Sphere

As suggested by Archimedes, if the radius of the cylinder, cone, and the sphere is "r" and they have the same cross-sectional area, their volumes are in the ratio of 1:2:3. Hence, the relation between the volume of sphere, volume of cone, and volume of cylinder is given as:

Volume of Cylinder = Volume of Cone + Volume of Sphere

⇒ Volume of Sphere = Volume of Cylinder - Volume of Cone
As we know, the volume of cylinder = πr²h and volume of cone = one-third of the volume of cylinder = (1/3)πr²h
The volume of Sphere = Volume of Cylinder - Volume of Cone
⇒ Volume of Sphere = πr²h - (1/3)πr²h = (2/3)πr²h
In this case, height of cylinder = diameter of sphere = 2r
Hence, volume of sphere is (2/3)πr²h = (2/3)πr²(2r) = (4/3)πr³

Volume of Sphere Formula

The volume of sphere formula can be given for a solid as well as the hollow sphere. In the case of a solid sphere, we only have one radius but in the case of a hollow sphere, there are two radii, having two different values of radius one for the outer sphere and one for the inner sphere.

Volume of Solid Sphere

If the radius of the sphere formed is r and the volume of the sphere is V. Then, the volume of the sphere is given by:
Volume of Sphere, V = (4/3)πr³

Volume of Hollow Sphere

If the radius of the outer sphere is R, the radius of the inner sphere is r and the volume of the sphere is V. Then, the volume of the sphere is given by:
Volume of Sphere, V = Volume of Outer Sphere - Volume of Inner Sphere = (4/3)πR³ - (4/3)πr³ = (4/3)π(R³- r³)

How to Calculate Volume of Sphere?

The volume of a sphere is the space occupied inside a sphere. The volume of the sphere can be calculated using the formula of the volume of sphere. The steps to calculate the volume of a sphere are:

Step 1: Check the value of the radius of the sphere.
Step 2: Take the cube of the radius.
Step 3: Multiply r³ by (4/3)π
Step 4: At last, add the units to the final answer.

Let us take an example to learn how to calculate the volume of sphere using its formula.

Example: Find the volume of sphere having a radius of 4 inches.
Solution: As we know, the volume of sphere, V = (4/3)πr³
Here, r = 4 inches
Thus, volume of sphere, V = (4/3)πr³ = ((4/3) × π × 4³) in³
⇒ V = 268.08 in³
∴ The volume of sphere is 268.08 in³.

Volume of Sphere Examples

Example 1: What is the amount of air that can be held by a spherical ball of diameter 14 inches?
Solution: We need to find the volume of the ball.
The radius of the ball will be half the diameter = 14/2 inches = 7 inches

Using the volume of sphere formula, the volume of the ball is
Volume of the ball = (4/3)πr³= ((4/3) × (22/7) × 7³) =1436.75 in³
∴ The amount of air that can be held by the spherical ball of diameter 14 inches is 1436.75 cubic inches.
Example 2: Maria has three wax marbles of radii 6 inches, 8 inches, and 10 inches. She melted all the marbles to recast them into a single solid marble. Can you find the radius of the resulting marble?
Solution: Let the radius of 3 marbles be \(r_{1}\), \(r_{2}\) and \(r_{3}\) and the radius of resulting marble is R. Tp calculate the value of R lets form the equation using the volume of sphere.
Volume of resulting marble = Volume of marble 1 + Volume of marble 2 + Volume of marble 3
(4/3)πR³= (4/3)π\((r_{1})\)³+(4/3)π\((r_{2})\)³ + (4/3)π\((r_{3})\)³
⇒ R³= ( \((r_{1}\))³+ (\(r_{2}\))³+ (\(r_{3}\))³
⇒ R³ = 6³ + 8³ + 10³ = 1728
⇒ R = 12 inches

∴ The radius of the resulting marble is 12 inches.

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Practice Questions on Volume of Sphere

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FAQs on Volume of Sphere

What is the Volume of Sphere?

The volume of sphere is the amount of air that a sphere can be held inside it. The formula for calculating the volume of a sphere with radius 'r' is given by the formula volume of sphere = (4/3)πr³.

How to Calculate the Volume of a Sphere?

We can calculate the volume of a sphere using the below-given steps:

Step 1: First find the value of radius or diameter.
Step 2: Use the formula of volume of sphere (4/3)πr³.
Step 3: Write the unit in the end, once the value is obtained.

☛ Check:

What is the Area and Volume of Sphere?

The surface area of a sphere is the total area or region covered by the surface of the sphere. The surface area of a sphere is given by two following formulas:

The surface area of sphere = 2πrh
If the diameter of the sphere = 2r, then the surface area of the sphere = 4πr² square units.

The area of a sphere is always expressed in square units, like m², in², cm², yd², etc.

The volume of a sphere is the total amount of capacity immersed in a sphere that can be calculated using the volume formula for the sphere which is V = (4/3)πr³. The volume of a sphere is always measured in cubic units.

☛ Check:

What is the Relation Between the Volume of Sphere and the Volume of Cylinder?

The relation between the volume of sphere and the volume of cylinder is that the volume of the sphere is two-third of the volume of the cylinder with a height equal to the diameter of the sphere and the same radius.

What is the Ratio of the Surface Area to the Volume of the Sphere of Unit Radius?

The formula of volume of a sphere = (4/3)πr³ and the formula of the surface area of sphere = 4πr². Hence, the ratio of surface area to the volume of the sphere of unit radius is ((4/3)π)/4π = 1:3

How Does the Volume of Sphere Change When the Radius of Sphere is Halved?

The volume of sphere gets one eighth when the radius is halved as, r = r/2. As, volume of sphere = (4/3)πr³ = (4/3)π(r/2)³ = (4/3)π(r³/8) = volume/8. Thus, the volume of sphere gets one eighth as soon as its radius gets halved.

How To Calculate the Volume of Sphere With Diameter?

The general formula for the volume of sphere in terms of its radius is given as V = (4/3) π r³. Let's say 'd' is its diameter, according to the definition of diameter, we have d = 2r. From this, we get the value of radius = (d/2). Substituting this in the volume of a sphere formula, the volume of a sphere in terms of diameter is represented as V = (πd³)/6.

☛ Check: Surface Area of a Sphere in Terms of Diameter

How Do You Measure the Volume of Sphere?

The formula for measuring the volume of a sphere is (4/3)πr³. We can simply measure the volume of any spherical shell by substituting the values of the parameters like radius and diameter in the volume formula.

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