The sum of the areas occupied by the object surface is the surface area. A three-dimensional shape is typically a solid with depth and height. A sphere, on the other hand, is a three-dimensional shape in which each point on its surface is equally spaced from the centre. The integration method can be used to compute the surface area of a sphere. The surface area of a sphere is equal to four times the product of \(\pi \left( {pi} \right)\) and the square of the radius. The size of the sphere, i.e. the radius of the sphere, determines the Surface Area of Sphere. The larger the radius, the larger the sphere’s surface area. The unit of the surface area of a sphere will be the square of any unit of length like \({\text{m}}{{\text{m}}^{\text{2}}}{\text{,}}\,{\text{c}}{{\text{m}}^{\text{2}}}{\text{,}}\,{{\text{m}}^{\text{2}}}\) etc.

Maths is a tough subject. It is important for students to learn these basic concepts in Mathematics. They can master the subject with regular practice. In this article, let’s understand everything about the surface area of a sphere in detail.

What do We Understand by a Sphere?

Let’s understand sphere with the help of an example. Consider a basketball. What is the shape of a basketball? Is it a circle? No. We cannot say that a basketball is a circle shape.

But there is a connection of circle in a basketball. How? Try rotating a circle along any of its diameters. Can you see what is coming out?

Isn’t that having a similar shape to that of a basketball? Yes, it is.  Basketball is a three-dimensional shape obtained while rotating a circle along any of its diameters. The basketball is of the shape of a sphere.

What is the Definition of a Sphere?

A circle is a two-dimensional shape, which can be easily drawn on a piece of paper. On the other hand, a sphere is a three-dimensional shape like the shape of a football or a basketball. Three coordinate axes, x-axis, y-axis and z-axis is 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 circumference of the sphere is called the radius of the sphere. The size of a sphere is determined by the radius of the sphere.

What are the Properties of a Sphere?

1. A sphere is a symmetrical three-dimensional figure.
2. All the points on the surface of a sphere are equidistant from the origin or the centre.
3. It has no vertex and no edges.
4. It has only one curved surface.

What are the Examples of a Sphere?

We can see a wide range of spherical objects all around us. Football, basketball, and other sports balls fall into this category.

What are the Types of a Sphere?

There are two types of spheres.

1. Solid sphere
2. Hollow sphere

Solid Sphere:

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

Hollow sphere:

A hollow sphere is a sphere that has only the outer spherical boundary and nothing is filled inside.

For easy understanding, consider a ball of ice cream.

What is the Surface Area?

The amount of outer space covering a three-dimensional shape is known as the surface area. The area of any three-dimensional geometric shape can be classified into three types. These are:

1. Curved surface area
2. Lateral surface area
3. Total surface area

Curved surface area

The curved surface area is the area of all the curved regions of the solid.

Lateral surface area

The lateral surface area is the area of all the faces except the top and bottom faces or bases.

Total surface area

The total surface area is the area of all the faces (including top, and bottom faces) of the solid object.

What do you Understand by the Surface Area of a Sphere?

A sphere is a three-dimensional geometrical shape that is perfectly round. There are so many spherical objects around us. Archimedes developed the surface area formula over two thousand years ago. 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 is the curved surface area of it as there is no difference between the curved surface area and the total surface area of a sphere.

The value of \(\pi \)

A constant term \(\pi \) (pi) is used in the formula of the area of a circle. \(\pi \) is a constant term, also known as Archimedes’ constant. One way to define π is that it is the ratio of the circumference of a circle to its diameter. It is an irrational number whose value is \(3.141592653589793238.\) For the common use in practice, the value of \(\pi \) is approximately taken as \(3.14\) when used as a decimal number and is taken as \(\frac{{22}}{7}\) when used as a fraction .

What is the Formula of the Surface Area of a Sphere?

(a) Surface area of a sphere formula when the radius of the sphere is given

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

(b) Surface area of a sphere formula when the diameter of a sphere is given

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

What is a Hemisphere?

Take a lemon that is spherical and cut it into two halves.

The two lemon pieces which we have now are nothing but two hemispheres. A hemisphere is an exact half of a sphere. In essence, two identical hemispheres make a sphere. Similarly, we can take an example of a half watermelon.

What is the Formula of the Surface Area of a 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
2. Total surface area of a hemisphere

Total surface area of a hemisphere

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

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

Total Surface Area of a Hemisphere

The total surface area includes the circular surface and the curved surface area of the hemisphere.
The total surface area of a hemisphere is \(A = 2\pi {r^2} + \pi {r^2} = 3\pi {r^2}\,{\text{unit}}{{\text{s}}^2}\)
where \(r\) is the radius of the hemisphere.

What are the Factors Affecting the Surface Area of a Sphere?

The surface area of the sphere is determined by the size of the sphere. The size is based on the radius of the sphere. The more the radius, the more will be the surface area of a sphere. The surface area of a sphere is given by \(A = 4\pi {r^2},\) where \(r\) is the radius of the sphere. In the formula for the surface area of a sphere, \(4\) and \(\pi \) are constants. The surface area of a sphere directly depends on the radius of the sphere. \(A \propto {r^2}\)
If we compare two spheres of different sizes, the sphere with a greater radius will have a larger surface area.

Here, the pink sphere is having a radius of \(1\,{\text{m,}}\) and the blue sphere is having a radius of \(2\,{\text{m}}{\text{.}}\) Since the blue sphere has a longer radius, the surface area of the blue sphere will be larger than the pink sphere.

What are the Units of the Surface Area of a Sphere?

The unit of the radius of a sphere can be any units of length like \({\text{mm}},\,{\text{cm}},\,{\text{m}},\) etc. Since the radius is squared in the formula of the surface area of a sphere, the unit should also be squared. Hence, the unit of the surface area of a sphere will be the square of any unit of length like \({\text{m}}{{\text{m}}^2},\,{\text{c}}{{\text{m}}^2},\,{{\text{m}}^2},\) etc.

Solved Examples – Surface Area of a Sphere

Q.1. Find the surface area of a sphere whose radius is \(7\,{\text{cm}}\) considering \(\pi = \frac{{22}}{7}.\)
Ans: Given the radius of a sphere, \(r = 7\,{\text{cm}}\)
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 \(7\,{\text{cm}} = 4\pi \times {7^2}\,{\text{c}}{{\text{m}}^2}\)
\( = 4 \times \frac{{22}}{7} \times 7 \times 7\,{\text{c}}{{\text{m}}^2}\) \( = 4 \times \frac{{22 \times 7 \times 7}}{7}\,{\text{c}}{{\text{m}}^2}\) \(= 616\,{\text{c}}{{\text{m}}^2}\)

Q.2. Find the surface area of a sphere whose diameter is \(10\,{\text{cm}}\) considering \(\pi = \frac{{22}}{7}.\)
Ans: Given, the diameter of a sphere \(10\,{\text{cm}}\)
So, the radius of the sphere \( = \frac{{{\text{diameter}}}}{2} = \frac{{10\,{\text{cm}}}}{2} = 5\,{\text{cm}}\)
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 \(5\,{\text{cm}} = 4 \times \frac{{22}}{7} \times 5 \times 5\,{\text{c}}{{\text{m}}^2}\)
\( = 4 \times \frac{{22 \times 5 \times 5}}{7}\,{\text{cm}}\)
\(= 314.28\,{\text{c}}{{\text{m}}^2}\)

Q.3. The surface area of a sphere is \(201\,{\text{sq}}\,{\text{m}}.\) Find the radius of the sphere considering \(\pi  = 3.14.\)
Ans: Given the surface area of a sphere \(= 201\,{\rm{sq}}\,{\rm{m}}\)
We know that the surface area of a sphere is calculated as: \(A = 4\pi {r^2}\)
We can find the radius of the sphere from the surface area of a sphere as \(r = \sqrt {\frac{A}{{4\pi }}} \)
So, the radius of the given sphere \(r = \sqrt {\frac{{201}}{{4 \times 3.14}}} \,{\rm{m}}\) \(= \sqrt {16.003\,} \,{\text{m}}\)
\(\approx 4\,{\text{m}}\)

Q.4. What is the total surface area of a solid hemispherical object of radius \(2\,{\text{cm}}\) 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 \(= 2\,{\text{cm}}\)
So, the surface area of the object \( = 3 \times \frac{{22}}{7} \times 2\, \times 2{\text{c}}{{\text{m}}^2}\)
\( = 3 \times \frac{{22 \times 2\, \times 2}}{7}{\text{c}}{{\text{m}}^2}\)
\(= 37.71\,{\text{c}}{{\text{m}}^2}\)

Q.5. What is the curved surface area of a hemisphere if the diameter is \(12\,{\text{cm}}\)
Ans: The diameter is \(12\,{\text{cm}}{\text{.}}\)
The radius is \(\frac{{12}}{2}\,{\text{cm=6}}\,{\text{cm}}\)
The curved surface area of the hemisphere \( = 2\pi {r^2} = 2 \times \frac{{22}}{7}\, \times 6 \times {\text{6}}\,{\text{c}}{{\text{m}}^2}\)
\(= 226.28\,{\text{c}}{{\text{m}}^2}\)

Frequently Asked Questions

We have provided some frequently asked questions about surface of sphere here:

Q.1. What is the formula of the surface area of a sphere?
Ans: The formula of the surface area of a sphere is \(4\pi {r^2}\)

Q.2. Why the formula for the surface area of a sphere is \(4\pi {r^2}?\)
Ans: The Greek mathematician Archimedes discovered that the total surface area of a sphere is the same as the lateral surface area of a cylinder that has the same radius as the sphere and a height that is equal to the diameter of the sphere.
The lateral surface area of the cylinder is \(2\pi rh\) where \(h = 2r\)
The lateral surface area of the cylinder \( = 2\pi r\left({2 r} \right) = 4\pi {r^2}\)
Therefore, the surface area of a sphere with radius \(r\) equals \(4\pi {r^2}\).
This can also be derived using integral calculus.

Q.3. What are the CSA and TSA of a sphere?
Ans: The curved surface area (CSA) and the total surface area(TSA) are the same for a sphere as a sphere has no flat surface.

Q.4. What is the volume and surface area of a sphere?
Ans: The volume of a sphere is the amount of space contained by it. The formula of volume of the sphere \(\frac{4}{3}\pi {r^3}\)
The amount of outer space covering a three-dimensional shape is known as the surface area.
The formula of the surface area of the sphere \(4\pi {r^2}\).

Q.5. How do you find the surface area of a sphere?
Ans: If we know the radius of the sphere, we can easily get the surface area of it using the formula \(4\pi {r^2}\), where \(r\) is the radius.

Study At Embibe

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a. Class 8 Maths Practice Questions

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c. Class 10 Maths Practice Questions

d. Class 11-12 Maths Practice Questions

From this article, we have learned about Sphere & Hemisphere formula to find the surface area along with their examples. We hope this detailed article on the Surface area of the Sphere is helpful to you. If you have any questions, please reach us through the comment box below and we will get back to you as soon as possible.