Volume of Sphere: Formula, Derivation, Examples

What is the Volume of Sphere?

What is the volume of sphere? is a commonly asked question when we study 3D shapes in maths. The volume of sphere means the amount of space that is present inside a sphere. Just like air fills a football or water fills a round tank, the volume shows how much the sphere can hold.

A sphere is a perfectly round shape in three dimensions, just like a ball or a bubble. It has no corners, no edges, and no flat surfaces. Learning about the volume of sphere helps us solve many questions in exams as well as in real life related to storage, capacity, and measurement. So, keep reading to learn how to calculate volume of sphere using a formula.

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Unit of Volume of Sphere

The volume of sphere tells us the space inside it, so its unit is always written in cubic form. This means the unit is raised to the power of 3. For example, if the radius of a sphere is measured in centimeters, then the volume will be in cubic centimeters (cm³). If the radius is in meters, then the volume will be in cubic meters (m³).

In the metric system, common units are cm³ and m³. In the US system, the units can be cubic inches (in³) or cubic feet (ft³). The unit of volume of sphere depends on the radius given in the question. Additionally, there are two types of spheres: a solid sphere and a hollow sphere, with each having a different formula for calculating volume.

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Derivation of Volume of Sphere

Derivation of volume of sphere was first explained by Archimedes. He showed that if the radius of a cylinder, a cone, and a sphere is the same (r), and they have the same cross-sectional area, then their volumes are related in a special ratio of 1 : 2 : 3.

From this relation, we can write:

Volume of Cylinder = Volume of Cone + Volume of Sphere

So, the volume of sphere is:

Volume of Sphere = Volume of Cylinder - Volume of Cone

In this, we know:

• Volume of Cylinder = πr²h

• Volume of Cone = 1/3 πr²h

Now, putting values:

Volume of Sphere = πr²h - 1/3 πr²h = 2/3 πr²h

Here, the height of the cylinder is equal to the diameter of the sphere, h = 2r.

Volume of Sphere Formula = 2/3 πr² × 2r = 4/3 πr³

4/3 πr³ is the standard volume of sphere formula we use to calculate the space inside any sphere. Let’s now learn about the volume of solid sphere and the volume of hollow sphere.

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Volume of Solid Sphere

A solid sphere is a sphere that is completely filled and has only one radius. To find the volume of solid sphere, we use the radius of the sphere in the formula.

If the radius of the solid sphere is r, then its volume (V) is given by:

Volume of Solid Sphere, V = 4/3 × π × r³

Volume of Hollow Sphere

A hollow sphere is a sphere that is empty from the inside, like a hollow ball. It has two radii: the radius of the outer sphere (R) and the radius of the inner sphere (r).

The volume of hollow sphere is the space between the outer and inner surfaces. It is calculated as:

Volume of Hollow Sphere: V = Volume of Outer Sphere - Volume of Inner Sphere

• V = 4/3 × π × R³ - 4/3 × π × r³

• V = 4/3 × π × (R³ - r³)

Read More: Numerator and Denominator

How to Calculate Volume of Sphere?

After learning about the volume of sphere, it is also important for you to understand how to calculate volume of sphere in real questions using the formula. Here are the simple steps to calculate it:

• Step 1: Check the value of the radius of the sphere (r).

• Step 2: Take the cube of the radius; that is, calculate r³.

• Step 3: Multiply r³ by 4/3 × π.

• Step 4: Write the answer with the correct unit, like cm³ or m³.

By following these steps, you can easily calculate the volume of sphere, whether it's solid or hollow. To better understand how to calculate volume of sphere, check the examples below.

Volume of Sphere Examples With Solutions

Here are some easy-to-understand volume of Sphere examples to help you understand how to calculate the space inside a sphere.

Example 1: What is the volume of a spherical ball of radius 5 cm?

Solution:

We use the volume of sphere formula: V = 4/3 × π × r³

Volume = 4/3 × π × 5³ = 4/3 × π × 125 ≈ 523.598 cm³

Example 2: Find the volume of solid sphere whose diameter is 12 cm.

Solution:

Radius r = diameter ÷ 2 = 12 ÷ 2 = 6 cm

Volume of solid sphere = 4/3 × π × 6³ = 4/3 × π × 216 ≈ 904.78 cm³

Example 3: A hollow metal ball has an outer radius of 10 cm and an inner radius of 8 cm. Find the volume of metal used to make the ball.

Solution:

Volume of hollow sphere = 4/3 × π × (R³ - r³)
= 4/3 × π × (10³ - 8³)
= 4/3 × π × (1000 - 512)
= 4/3 × π × 488 ≈ 2044.77 cm³

Example 4: A toy manufacturer melts three spherical toys of radii 3 cm, 4 cm, and 5 cm to make a single new toy. Find the radius of the new toy.

Solution:

Volume of new toy = sum of volumes of old toys
4/3 × π × R³ = 4/3 × π × (3³ + 4³ + 5³)
R³ = 27 + 64 + 125 = 216
R = 6 cm

Also Read: 2 Digit Subtraction

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