Sphere: Properties and Examples

Sphere: A sphere is a three-dimensional geometric shape formed by joining infinite numbers of points equidistant from a central point. The radius of the sphere is the distance between a point on its surface and the centre of the sphere. The volume of a sphere is the space it takes upon its surface. A sphere is a critical geometric figure that is present across many areas such as construction, engineering, and many more. 

Solving problems related to spheres in Geometry requires us to derive the area of a sphere. For a sphere, the surface area can be calculated with the formula 4π r², whereas its volume can be calculated with (4/3)π r³, where r is the radius. In this article, we will learn terms related to spheres, like radius, the volume of sphere etc. 

Sphere Shape Definition

We all have seen a football, which is a circle, but football is a 3-D figure. How? Try rotating a circle along any of its diameters, and you will see a sphere coming out.

Sphere Area of Sphere: Formulas, Examples

Thus, the shape of a football can be obtained by rotating a circle along any of its diameters.

Is Sphere a Circle?

A circle is a two-dimensional shape that can be easily drawn on a piece of paper. On the other hand, a sphere is a three-dimensional shape, like a football or a basketball. The three coordinate axes, \(x\)-axis, \(y\)-axis and \(z\)-axis, are used to define the shape of a sphere.

Like a circle, the sphere also has a centre point. Every point on the surface of a sphere is equidistant from its centre. This fixed distance from the centre to any point on the sphere’s circumference is called the sphere’s radius. The radius of the sphere determines the size of a sphere.

Types of Spheres

Spheres can be categorised mainly into two types:

1. Solid Sphere

A solid sphere is a three-dimensional object in the form of the sphere and filled up with the material it is made up of. For example, planets are a type of solid sphere.

2. Hollow Sphere

A hollow sphere is a sphere that has only the outer spherical boundary, and nothing is filled inside. For easy understanding, consider the example of a basketball.

It is spherical but hollow from the inside. Thus, a basketball is a perfect example of a hollow sphere.

Sphere Shape

Any spherical object is a round solid or a hollow sphere, with every point on its surface equidistant from its centre. Moreover, a spherical object does not have any faces. In addition to this, a sphere does not have any flat surface or an edge or a vertex.

Examples of Sphere Shape

As already discussed, a sphere is a 3-D shape, round, and solid or hollow from inside with every point on its surface equidistant from the centre. Some sphere shape examples are:

As we can see from the above-given figures, all the given shapes are spherical. All the above-given figures have no vertex, no edge, and only one surface, which is a curved surface.

Sphere Diagram

A sphere is the set of all points in three-dimensional space equidistant from a single point. Let us have a look at a sphere in 3-D space where the three coordinate axes, \(x\)-axis, \(y\)-axis, and \(z\)-axis, are used to define the shape of a sphere.

The radius of a sphere has one endpoint on the sphere surface and the other endpoint at the centre of that sphere. The diameter of a sphere must contain the centre.

Let us understand with the help of the below-given figure.

Volume of Sphere and Formula

The volume of a sphere is the amount of space contained by it. Let us understand this with the help of an example, try poking a football and make a hole. Fill it with water to the brim. Now, pour the water into a beaker and measure how much water is contained in that. That is the volume of that football.

Archimedes’ principle states that a body that is totally or partially immersed in a fluid is subject to an upward force equal in magnitude to the weight of the fluid it displaces. In simple words, if we drop a sphere into a container filled with water, then the amount of water displaced equals the volume of the sphere.

Formulas of Volume of Sphere

The volume of a sphere formula is when the radius is given and when the diameter is given.

1. Volume of a sphere when the radius of the sphere is given:

The volume of a sphere can be calculated using the formula \(V = \frac{4}{3}\pi {r^3}\) cubic units, where \(r\) is the radius of the sphere.

2. Volume of a sphere when the diameter of a sphere is given:

The volume of a sphere can be calculated using the formula \(V = \frac{4}{3}\pi {\left( {\frac{d}{2}} \right)^3}\) cubic units, where \(d\) is the diameter of the sphere.

The unit of the radius of a sphere can be any units of length like \({\rm{mm}},\,{\rm{cm}},\,{\rm{m}}\), etc. Since the radius is cubed in the formula of volume of a sphere, the unit should also be cubed. Hence, the unit of volume of a sphere will be the cube of any unit of length like \({\rm{m}}{{\rm{m}}^{\rm{3}}}{\rm{,c}}{{\rm{m}}^{\rm{3}}}{\rm{,}}{{\rm{m}}^{\rm{3}}}\), etc.

Surface Area and Formula of a Sphere

As we have already learned that a sphere is a perfectly round 3-D geometrical shape. The formula of surface area was introduced by Archimedes. The surface area of the sphere is described as the number of square units needed to cover the surface of the sphere. The surface area of a sphere defines the curved surface area of it as there is no difference between the curved surface area and the total surface area of a sphere as it has no flat surfaces.

Formula of Surface Area of Sphere

1. The surface area of a sphere when the radius of the sphere is given:

The surface area of a sphere can be calculated using the formula,
\(A = 4\pi {r^2}\) square units,
where \(r\) is the radius of the sphere.

2. The surface area of a sphere when the diameter of a sphere is given:

The surface area of a sphere is \(A = 4\pi {\left( {\frac{d}{2}} \right)^2}{\rm{square}}\,{\rm{units}}\)
where \(d\) is the diameter of the sphere.

What is a Hemisphere?

The prefix ‘hemi’ means half, and a sphere, as already learned, is a 3-D shape that is perfectly round. So, a hemisphere is an exact half of a sphere. In essence, two identical hemispheres make a sphere.

Formula for Surface Area of Hemisphere

A hemisphere is a three-dimensional shape and exactly half of a sphere. The surface area of the hemisphere can be classified into two types.

1. Curved Surface Area of a Hemisphere:

The curved surface of the hemisphere will also be exactly half of the surface area of a sphere as it does not include the circular base.

The curved surface area of a hemisphere is \(A = 2\pi {r^2}\) square units,
where \(r\) is the radius of the hemisphere.

2. Total Surface Area of a Hemisphere:

The total surface area includes the circular base and the curved surface area of the hemisphere.
\(A = \left({2\pi {r^2} + \pi {r^2}} \right) = 3\pi {r^2}\) square units.

Solved Examples on Sphere

Q.1. Find the volume of a sphere whose radius is \(7\,{\rm{cm}}\) considering \(\pi  = \frac{{22}}{7}\).
Ans:
Given the radius of a sphere, \({\rm{r = 7cm}}\)
We know that the volume of a sphere is calculated as
\(V = \frac{4}{3}\pi {r^3}\)
So, the volume of a sphere of radius  \(7\;{\rm{cm}} = \frac{4}{3}\pi \times {7^3}\;{\rm{c}}{{\rm{m}}^{\rm{3}}}\)
\(= \frac{4}{3} \times \frac{{22}}{7} \times 7 \times 7 \times 7\;{\rm{c}}{{\rm{m}}^{\rm{3}}}{\rm{\;}}\;\)
\(= \frac{{4 \times 22 \times 7 \times 7}}{3}{\rm{\;c}}{{\rm{m}}^3} = \frac{{4312}}{3}\;{\rm{c}}{{\rm{m}}^3}\)
\( = 1437.33\;{\rm{c}}{{\rm{m}}^3}\)
Hence, the volume of the sphere with a radius \(7\;{\rm{cm}}\) is \(1437.33\;{\rm{c}}{{\rm{m}}^3}.\)

Q.2. Find the surface area of a sphere whose radius is \(10\;{\rm{cm}}\) considering \(\pi = \frac{{22}}{7}.\)
Ans:
Given, the radius of a sphere \( = 10\;{\rm{cm}}\)
We know that the surface area of a sphere is calculated as
\(A = 4\pi {r^2}\)
So, the surface area of a sphere of radius \(10\;{\rm{cm}} = 4 \times \frac{{22}}{7} \times 10 \times 10\;{\rm{c}}{{\rm{m}}^2}\)
\( = \frac{{4 \times 22 \times 10 \times 10}}{7}\;{\rm{c}}{{\rm{m}}^3}\)
\( = 1257.14\;{\rm{c}}{{\rm{m}}^2}\)

Q.3. What is the total surface area of a solid hemispherical object of radius \(5\;{\rm{cm}}\) considering \(\pi = \frac{{22}}{7}.\)
Ans:
We know that the surface area of a solid hemisphere is calculated as
\(A = 3\pi {r^2}\)
Given, the radius of the object \( = 5\;{\rm{cm}}\)
So, the total surface area of the object \(\; = 3 \times \frac{{22}}{7} \times 5 \times 5\;{\rm{c}}{{\rm{m}}^2}\)
\( = \frac{{3 \times 22 \times 5 \times 5}}{7}\;{\rm{c}}{{\rm{m}}^2}\)
\( = 235.71\;{\rm{c}}{{\rm{m}}^2}\)

Q.4. What is the curved surface area of a hemisphere if the diameter is \(6\;{\rm{cm}}.\)
Ans: The diameter is \(6\;{\rm{cm}}.\)
The radius is \(\frac{6}{2}\;{\rm{cm}} = 3\;{\rm{cm}}\)
The curved surface area of the hemisphere \( = 2\pi {r^2} = 2 \times \frac{{22}}{7} \times 3 \times 3\;{\rm{c}}{{\rm{m}}^2}\)
\( = 56.57\;{\rm{c}}{{\rm{m}}^2}\)

Q.5. Rachana has two wax marbles of radii \(\;8\;{\rm{cm}}\) and \(10\;{\rm{cm}}.\) He melted all two marbles and recast them into a single solid marble. Find the radius of the new single solid marble?
Ans: The marbles are in the shape of a sphere. So, when two or more marbles are melted and recast into another big marble, their volume will remain the same.
Therefore, volume of the new single solid marble = Volume of marble of \(8\;{\rm{cm}} + \;\) Volume of marble of \(10\;{\rm{cm}}\)
We know that the volume of a sphere is calculated as
\(V = \frac{4}{3}\pi {r^3}\)
Considering the radius of the single solid marble as \(R\), we get the volume as
\(\frac{4}{3}\pi {R^3} = \frac{4}{3}\pi  \times {8^3} + \frac{4}{3}\pi  \times {10^3}\)
\( \Rightarrow {R^3} = {8^3} + {10^3}\)
\( \Rightarrow {R^3} = 1512\)
\( \Rightarrow R = \sqrt[3]{{1512}} = 11.48\)
Hence, the radius of the single solid marble is \(11.48\;{\rm{cm}}.\)

Related Topics to Study

Summary

In this article, we learned about the shape of a sphere. We also learned about the volume of a sphere and the difference between a circle and a sphere. We also learned about the sphere types, i.e., solid sphere and hollow sphere.

FAQs About Sphere

Let us look at some of the commonly asked questions about Sphere:

Q.1. What is the smallest unit in geometry?
Ans: The smallest unit in the geometry is a point.

Q.2. How do you describe a sphere?
Ans: Sphere shape definition can be described as a \(3-D\) shape that is round and solid or hollow from inside with every point on its surface equidistant from the centre. A sphere has a surface area and volume based on its radius.

Q.3. What does sphere mean?
Ans: A sphere is the set of all points in three-dimensional space that are equidistant from a single point.

Q.4. What is a shape of a sphere?
Ans: A sphere is a geometrical object in \(3-D\) which is of a round shape.

Q.5. What is special about the sphere?
Ans: A sphere is perfectly symmetrical around its centre.

Q.6. How many sides a sphere has?
Ans: A sphere does not have any sides since it is a round-shaped object. It has a curved surface and not a flat surface.

Now that you are provided with all the necessary information on spheres, we hope this detailed article is helpful to you as soon as possible. If you have any queries on this article or in general about spheres, ping us through the comment box below, and we will get back to you as soon as possible.