Volume of a Sphere

We know that the sphere is generally defined as a solid three-dimensional round structure in which every point on its surface is measured to be equidistant from its centre. So we can say that the sphere is technically a three-dimensional shape or a structure that has no edges or vertices. Then, the volume of a sphere is the determination of the empty space it can occupy. In other words, the volume of a sphere is its total capacity. The shape of the sphere is usually round and it is a three-dimensional structure, for example, a football is a sphere. A sphere will have three axes x-axis, y-axis and z-axis and these axes define the shape of the sphere.

In this article, we will learn how to find the volume of a sphere using the volume of a sphere formula, surface areas of a sphere and sphere formula along with a few solved numerical problems for boosting our understanding. 

Volume of a Sphere

So, now the question that arose is what is the volume of a sphere? 

The volume of a sphere can be defined as the measurement of the total available space that can be occupied by a sphere structure. Suppose that we draw a circle on a plain sheet of paper, use a circular disc and paste a string along its diameter and rotate it in the direction of the string. This will result in the formation of the shape of a sphere. 

The volume of a sphere formula will be derived depending on the structure of the sphere, because some of the spheres will be solid such as billiards and some of the sphere will hollow such as football. But, the volume of the sphere technically depends on the diameter or the radius of the sphere, because if we consider the cross-section of the sphere, it will be a circle of a fixed radius, say r. 

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Now let us derive the volume of the sphere formula. We know that, according to Archimedes, if the radius of any solid geometrical structure such as a cylinder, cone, and the sphere is r and it is noticeable that all these structures are having the same cross-sectional area, their volumes are in the ratio of 1:2:3. Hence, the relation between the volume of a sphere, the volume of the cone and the volume of a cylinder can be written as:

\[\Rightarrow \text {Volume of Cylinder} = \text {Volume of a cone} + \text {Volume of Sphere} \]

\[ \Rightarrow \text{Volume of Sphere} = \text{Volume of Cylinder} - \text{Volume of Cone} \]........(1)

Now, we know that:

The volume of the cylinder = \[\pi r^{2} h \]

The volume of the cone = \[\frac{1}{3} \pi r^{2} h \]

Substituting these values in equation (1) we get:

\[ \Rightarrow \text{Volume of Sphere} = V = \pi r^{2} h - \frac{1}{3}\pi r^{2}h \]

\[\Rightarrow V = \frac{2}{3} \pi r^{2}h\] ……(2)

Where,

\[h\] - The height of the cylinder

We know that the height of the cylinder is equal to the diameter of the sphere. Therefore, equation (2) is given by:

\[\Rightarrow V = \frac{2}{3} \pi r^{2} (2r) \]

\[ \Rightarrow V = \frac{4}{3} \pi r^{3}\] ………(3)

Where, 

\[r\] - The radius of sphere formula

Equation (3) is known as the volume of a sphere formula.

The formula of volume of a sphere can be further derived for a solid as well as a hollow sphere. When we are considering a solid sphere, we only have one fixed radius, whereas, in the case of a hollow sphere, there are two radii, having two different numerical values of radius. The value of the outer radius will be different from the value of the inner radius. Thus, let us have a look at the volume of a sphere formula individually for a hollow sphere and for a solid sphere.

The volume of Solid Sphere:

Let us assume that we have a solid sphere of radius r then the volume of the solid sphere is given by:

\[ \Rightarrow V = \frac{4}{3} \pi r^{3}\]

The volume of Hollow Sphere:

Let us assume that we have a hollow sphere of radius inner radius r and R be the outer radius of the sphere, then the volume of the hollow sphere is given by:

\[\Rightarrow V = \frac{4}{3} \pi (R^{3} - r^{3}) \]

Similarly, we have tsa of hemisphere formula, csa of hemisphere formula which are the important formulae to be studied under the volume of a sphere formula concept.

Surface Areas of a Sphere

The surface areas of a sphere are the area or particular region of the outer surface of the given sphere. Thus, the surface areas of a sphere can be defined as the space covered by its outer surface (or curved surface area of a sphere) in three dimensions.  We know that the sphere is a three-dimensional solid having a definite circular shape, and the size of the sphere depends on the radius of the sphere. Therefore the formula of the total surface areas of a sphere is given by:

\[\Rightarrow A = 4 \pi r^{2} \] square units

Where,

\[r\] - The radius of the given sphere

Examples:

1. What is the volume of the air that can be held by a spherical ball of diameter 4cm?

Sol:

Given,

The diameter of the spherical ball = d = 4 cm

Then, the radius of the spherical ball = r = \[\frac{d}{2}\] = 2 cm

We are asked to determine the volume of the air held by the given ball. We know that the volume of the spherical ball is given by:

\[\Rightarrow V = \frac{4}{3} \pi r^{3} \].....(1)

Where, 

\[r\] - The radius of the sphere

Substituting the value of radius in equation (1) and on simplification we get:

\[\Rightarrow V = \frac{4}{3} \pi r^{3}\]

\[\Rightarrow V = \frac{4}{3} \pi (2)^{3} = \frac{32}{3} \pi cm^{3}\]

\[ \Rightarrow V = 33.51 cm^{3} \]

Therefore, the volume of the air held by the given ball is \[33.51 cm^{3} \].

2. Calculate the volume of the sphere whose diameter is 12 cm.

Sol:

Given,

The diameter of the spherical ball = d = 12 cm

Then, the radius of the spherical ball = r = \[\frac{d}{2} \] = 6 cm 

We are asked to determine the volume of the sphere. We know that the volume of the sphere is given by:

\[ \Rightarrow V = \frac{4}{3} \pi r^{3}\] …..(1)

Where, 

\[r\] - The radius of the sphere

Substituting the value of radius in equation (1) and on simplification we get:

\[\Rightarrow V = \frac{4}{3} \pi r^{3}\]

\[\Rightarrow V = \frac{4}{3} \pi (6)^{3} = \frac{864}{3} \pi cm^{3} \]

\[\Rightarrow V = 904.77cm^{3} \thickapprox 905 cm^{3} \]

Therefore, the volume of the sphere is \[905 cm^{3} \].