Surface Area of a Sphere: Formula, Derivation

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The surface area of a sphere is the area covered by the outer surface of the sphere. The surface area of an object is equal to the total of the areas covered by its surface. An object with depth and height forms a three-dimensional shape. A sphere is a three-dimensional form with evenly distributed points on its area. A sphere's surface area can be calculated using the integration method.

Table of Content

What is the Surface Area of a Sphere?
Formula of Surface Area of Sphere
Derivation of Surface Area of Sphere
How to Find the Surface Area of Sphere
Curved Surface Area of Sphere
Lateral Surface Area of Sphere
Total Surface Area of Sphere
Things to Remember
Sample Questions

Key Terms: Radius, Diameter, Sphere, Circle, Area, Volume, Lateral Surface Area, Curved Surface Area, Cylinder

What is the Surface Area of a Sphere?

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A sphere is an object where each of the points on its round surface is equal to the sphere's fixed centre. The radius is the distance between the centre and the surface. The area occupied by the curved surface of a sphere is known as its surface area. Both circle and sphere are round in shape. The key difference between the two is that the sphere is a three dimensional figure as it has volume whereas the circle is a two dimensional figure. The unit of surface area of the sphere is square units.

Sphere

Sphere

Formula of Surface Area of a Sphere

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A sphere is a three-dimensional object that resembles a circle and has a round shape. The surface area of every three-dimensional object can be divided into three categories. It's as follows:

Curved Surface Area
Lateral Surface Area
Total Surface Area

The formula for calculating the surface area of a sphere can be used to compute the sphere's surface area. Let us assume a sphere of radius ‘r’. The area ‘A’ of sphere can then be determined by the formula:

A = 4πr²

Derivation of Surface Area of a Sphere

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Because a sphere has a circular shape, we connect it to a parabola, such as a cylinder, to calculate its surface area. A cylinder is a shape with a curving surface having flat surfaces on both sides. When the radius of a cylinder and the radius of a sphere are the same, the sphere can precisely fit into the cylinder. This signifies that the cylinder's height is the same as the sphere's diameter. As a result, Archimedes showed that when the radius of a cylinder and sphere is 'r,' the surface area of a sphere is equal to the lateral area of a cylinder.

Thus,

Surface Area of a Sphere = Lateral Surface Area of a Cylinder

Here,

Lateral Surface Area of a Cylinder = 2πrh

where, r = radius of a cylinder and h = height of a cylinder

Now, we assume that sphere get perfectly fit in cylinder so the height of cylinder is also the diameter of cylinder

Height of Cylinder = Diameter of Sphere = 2r

So,

Surface Area of a Sphere = 2πrh

Now, In the surface area of a sphere we can replace h with 2r (because Height of cylinder = diameter of Sphere = 2r )

So, the surface area of a sphere = 2πrh = 2πr(2r) = 4πr²

Hence,

A = 4πr²

Here, r is the radius of the sphere.

How to Find the Surface Area of a Sphere

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Let’s take a look at this example to see how to use the formula to determine the surface area of a sphere.

Example: Find the surface area of a sphere with a radius of 6 cm.

Step 1: Note the given radius of the sphere. Here, the radius of the sphere is 6 cm.

Step 2: Now, we know that the surface area of sphere = 4πr², so by substituting the values in given formula we get, 4 × 3.14 × 6 × 6 = 452.16

Step 3: Thus, the surface area of a sphere is 452.16 cm2

Curved Surface Area of a Sphere

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The area of all the bent portions of a solid is called the curved surface area. Because a sphere has only one curved surface, its curved surface area equals its overall surface area. So,

Curved Surface Area of Sphere = Total Surface Area of Sphere = 4πr²

Lateral Surface Area of a Sphere

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The lateral surface area includes all regions excluding the bases (i.e., top and bottom).

We don't require any precise top or bottom portions in the shape of a sphere. So, the lateral Surface area of a sphere is the same as the total surface area of a sphere. Thus,

Lateral Surface Area of Sphere = Total Surface Area of Sphere = 4πr²

Total Surface Area of Sphere

The total surface area of a solid is the sum of its sides, top, and bottom. A sphere does not have any sides. So, it is calculated as 4πr2.

Total Surface Area of Sphere = 4πr²

Things to Remember

A sphere is an object where each of the points on its round surface is equal to the sphere’s fixed centre.
The total surface area, lateral surface area and curved surface area of a sphere are equal, i.e, 4πr².
The volume of a sphere of radius ‘r’ can be determined by the formula, V = 4/3 πr3
The radius of a sphere is the distance between the sphere's centre and any point on the sphere's surface. It is denoted by ‘r’. So, r = d/2.
The diameter of a sphere is the longest line in the centre of the sphere that touches both the ends of the sphere. Diameter is double of the radius of the sphere. It is denoted by ‘d’. So, d = 2r.

Sample Questions

Ques. If the surface area of a sphere is 314 cm² then find the diameter of the sphere. (2 marks)

Ans. Given,

The surface area of a sphere = 314 cm²

Hence, 4πr² = 314

r² = 314/4π

r² = 314/(4 × 3.14)

r² = 314/12.56

r² = 25

r = 5

r = d/2

So, d = 10

Thus, the diameter of a sphere is 10 cm.

Ques. Calculate the cost of painting a football that is shaped like a spherical with a radius of 7 cm. If the cost of football painting is INR 2.5 per square cm. (Assume π = 22/7) (2 marks)

Ans. We know that,

Surface area of a sphere = 4πr²

So, 4 × 22/7 × 7 × 7 = 616 cm 2

So, the area of football is 616 cm²

Now, the cost of painting football is 2.5 × 616 = INR 1540

Ques. How many square kilometres is the land area if three-fourths of the world’s surface is covered by water, assuming the earth is spherical with a radius of 6370 km? (2 marks)

Ans. Given that,

Radius of earth = 6370 km

Water on the earth = ¾ % of total area

Area of land = ¼ × 4πr2 = πr²

= 4 × 22/7 × (6370)²

= 127527400 km²

Ques. Find the surface area of a sphere with a diameter of 8 cm. (2 marks)

Ans. Given,

The diameter of sphere = 8 cm

So, radius = 8 /2 = 4 cm

We know that,

The surface area of a sphere = 4πr²

= 4 × 22/7 × 4 × 4

= 201.14 cm²

Ques. Calculate the volume of a sphere with 154 cm² of surface area. (3 marks)

Ans. Given,

The surface area of sphere = 154 cm²

We know that,

The surface area of a sphere = 4πr²

So, 4πr² = 154 cm²

4 × 22/7 × r² = 154

r² = 49/4

r = 3.5

Now, volume of a sphere = 4/3πr³

= 4 × 22/7 × 3.5 × 3.5 × 3.5

= 179.66 cm3

So, the volume of a sphere is 179.66 cm³.

Ques. One ladoo with a radius of 5 cm belongs to a shopkeeper. How many ladoos of radius 2.5 cm can be made from the same material? (3 marks)

Ans. Volume of laddoo having radius 5 cm (V₁) = 4/3 × 22/7 × (5)³

= 11000/21 cm³

Also, Volume of laddoo having radius 2.5 cm (V2) = 4/3πr³

= 4/3 × 22/7 × (2.5)³ cm³

= 1375/21 cm³

Therefore,

Number of laddoos of radius 2.5 cm that can be made = V₁/V₂ = 11000/1375 = 8

Ques. The diameter of the moon is about a quarter of the diameter of the earth. Calculate their surface area ratio. (3 marks)

Ans. Diameter of moon = 1/4 of diameter of earth

Let radius of earth per km

Then the radius of moon = (r/4) km

Now, surface area of earth = 4πr²

Surface area of moon = 4π(r/4)²

= 4π × (1/16) r² = (1/4) × πr²

Ratio between surface area of moon and earth = (1/4) × πr² : 4πr² = (1/4) : 4 = 1/16

Ques. As air is pumped into a spherical balloon, its radius rises from 7 cm to 14 cm. Calculate the ratio of the balloon's surface areas in both scenarios. (4 marks)

Ans. Radius of the spherical balloon initially, r1 = 7 cm

Surface area of the balloon initially = 4πr²

= 4 × π × (7)²

= 4 × π × 49

Radius of the spherical balloon after air is pumped, r² = 14 cm

Surface area of the balloon after air is pumped = 4πr2

= 4 × π × (14)²

= 4 × π × 196

Ratio of the two scenarios = Surface area of the balloon before air is pumped / Surface area of the balloon after air is pumped

= 4 × π × 49 : 4 x π × 196

= 1:4

Ratio of surface areas of the balloon in the two scenarios = 1 : 4

Ques. A sphere can be perfectly fit in the cylinder with a height of 10 cm. Find the surface area of a sphere? (3 marks)

Ans. Given,

The height of a cylinder = 10 cm

We know that,

Height of a cylinder = diameter of a Sphere = 2r

So, the radius of a sphere = 10/2 = 5 cm

Now,

Surface area of a sphere = 4πr2

= 4 × 22/7 × 5 × 5

= 314.28 cm is the surface area of the sphere.

Ques. The price per square metre of velvet is Rs 10. Calculate the price of 1000 rubber balls with a radius of 0.12 m. (2 marks)

Ans. The surface area of a ball = 4πr²

= 4 × 3.14 × 0.12 × 0.12

= 0.181 m

Now, the price of manufacturing ball = 0.181 × 10

= Rs 1.81

Now, the price of manufacturing 1000 balls = 1.81 × 1000 = Rs 1810

Read More:

Area of Hollow Cylinder	Surface Area of a Cylinder Formula	Surface Areas and Volumes MCQs
Surface Area Of A Hemisphere	Cube Formula	Mensuration Formula
Area of Prism	Cuboid	Surface Area of Right Circular Cone
Surface Area of Cone	Surface Area of Cuboid	Properties of Cylinder
Arc Length Formula	Geometry Formula	Introduction to Arithmetic Progression

Surface Area of a Sphere: Formula, Derivation

What is the Surface Area of a Sphere?

Formula of Surface Area of a Sphere

Derivation of Surface Area of a Sphere

How to Find the Surface Area of a Sphere

Curved Surface Area of a Sphere

Lateral Surface Area of a Sphere

Total Surface Area of Sphere

Things to Remember

Sample Questions

CBSE X Related Questions

1.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

2.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

3.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

4.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

5.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

6.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

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Surface Area of a Sphere: Formula, Derivation

What is the Surface Area of a Sphere?

Formula of Surface Area of a Sphere

Derivation of Surface Area of a Sphere

How to Find the Surface Area of a Sphere

Curved Surface Area of a Sphere

Lateral Surface Area of a Sphere

Total Surface Area of Sphere

Things to Remember

Sample Questions

CBSE X Related Questions

1.Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

2.An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

3.Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

4.If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

5.PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

6.Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

2.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

3.
Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

4.
If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

5.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

6.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$