What is the unit for the surface area of a sphere?

The unit for the surface area is square units, such as cm² or m² etc.

Is the surface area of a sphere dependent on its volume?

Yes, the surface area and volume depend on the cube and square of the radius, respectively, but both are independent quantities.

What happens to the surface area of a sphere if its volume is halved?

The reduction in the volume of a sphere decreases its radius, and the surface area will be reduced by a factor of the square of the cube root of 2.

Can the surface area of a sphere be negative in any case?

No, the surface area of a sphere is always a positive value.

Frequently Asked Questions

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Surface Area of a Sphere

1.0Master Spherical Geometry in Minutes

Learn how to calculate both the outer surface space and the total internal capacity of a perfectly round three-dimensional object. Understand the fundamental formulas, see how spheres relate to cylinders, and master high-yield board exam questions.

Class: 10 Mathematics (CBSE)
Chapter: Surface Areas and Volumes
Estimated Learning Time: 20–25 Minutes

2.0Learning Outcomes

After completing this lesson, you will be able to:

Define a sphere and understand its unique single-variable property.
State and apply the formula for the Surface Area of a complete sphere.
State and apply the formula for the Volume of a complete sphere.
Differentiate between two-dimensional area measurements and three-dimensional capacity measurements.
Solve board exam problems involving the melting, recasting, and combination of spherical solids.

3.0Area of Sphere

From the smallest cricket ball to the largest celestial body, spheres are all around us and understanding the surfaces related to the sphere is important for solving many real-life applications. The area of a sphere is one such application. The total surface area of a sphere is the area covered by the sphere in a three-dimensional setting.

4.0Understanding the Sphere

The sphere is nothing but a three-dimensional circle in which all the points on the surface lie at equal distances from the centre. In other words, it is the group of certain points at the same distance from the centre. The distance, like a circle, from the centre to the outer surface is known as the radius of the sphere, and the line joining two opposite sides of the sphere passing through the centre is known as the diameter of the sphere.

Diameter= $2 \times R a d i u s$

5.0What is the Formula for Sphere Area?

A sphere is the only three-dimensional figure having the same or equal curved and total surface area. The curved surface area of any 3-D figure is the area of the curved part only, while the total surface area includes the curved surface area along with the area of the base and/or the top part of the figure.

$Surface Area of the sphere(A) = 4 (π) r^{2}$

“The unit for the surface area of the sphere is the same as the square of the sphere's radius while solving the question.”

Area of a Sphere in Diameter

If, while solving any question, the diameter (d) of the sphere is given instead of the radius (r) of the sphere, then the formula for the area of the sphere can be expressed by using the relationship, $r = \frac{d}{2}$ , as:

$Surface Area of the sphere(A) = 4 π (\frac{d}{2})^{2}$

$A = 4 π \times \frac{d ^{2}}{4} = π d^{2}$

Steps to Use the Formula

Step 1: Determine the radius of the sphere and convert the unit into the required unit if needed.

Step 2: Solve the question using the correct values of each component. Always use $π = \frac{22}{7}$ if not mentioned otherwise.

6.0The Surface Area of the Sphere Proof

We can find the area of a sphere derivation by comparing the sphere with the lateral or curved surface area of a cylinder, which is also a 3-D figure. The derivation for the formula is:

Step 1: Comparing the sphere to a cylinder

Consider a sphere fitting perfectly within a cylinder with the same radius(r) as the cylinder and height (h) of the cylinder with the same as the diameter of the sphere such that h = 2r

Step 2: Lateral surface area of a cylinder

Lateral surface area of cylinder = $2 (π) r h$

We know h = 2r

So, Lateral surface area of cylinder = $2 (π) r (2 r)$

The new lateral surface area of a cylinder = $4 (π) r^{2}$

Since the surface area of the sphere = the lateral surface area of the cylinder

Hence,

The surface area of the sphere = $4 (π) r^{2}$

7.0Solved Examples

Problem 1: The surface area of a spherical tank is 1540 cm². Find the radius of the tank.

Solution: It is given that A = 1540cm²

$Surface Area of the sphere(A) = 4 (π) r^{2}$

$1540 = 4 \times \frac{22}{7} \times r^{2}$

$r^{2} = \frac{7 \times 1540}{22 \times 4}$

$r = \frac{7 \times 70}{4} = 3.5 10 c m^{2}$

Problem 2: The outer radius of a hollow spherical shell is 14 cm, and the inner radius is 7 cm. Calculate the surface area of the outer surface and the inner surface of the shell.

Solution: Given that,

Outer radius (R) = 14cm

Inner radius (r) = 7cm

The surface area of the outer surface = $4 π R^{2} = 4 (\frac{22}{7}) 1 4^{2}$

$= 4 \times 22 \times 28 = 2464 c m^{2}$

The surface area of the inner surface $4 π r^{2} = 4 (\frac{22}{7}) 7^{2}$

$= 4 \times 22 \times 7 = 616 c m^{2}$

Problem 3: The volume of a sphere is given as 904.32 cm³. Find the surface area of the sphere.

Solution: We know the Volume of the sphere (V) = $\frac{4}{3} π r^{3}$

$\frac{4}{3} π r^{3} = 904.32$

$r^{3} = \frac{904.32 \times 3}{4 π} \approx 216 c m^{3}$

$r = 3216 = 6 c m$

Now, using Surface Area of the sphere(A) = $4 π r^{2}$

$A = 4 \times \frac{22}{7} \times 6^{2} = \frac{3168}{7}$

$A = 452.38 c m^{2}$

8.0EUREKA by ALLEN – Learn Better, Score Highe

EUREKA by ALLEN is designed to simplify, enrich, and enhance your experience in Class 10. Through the use of fun and engaging video lessons, regular practice tests, and immediate help for any doubts you may have regarding the material; students have a firm understanding of the concepts they are studying and feel confident in their preparation for their board exams. No matter if you are attempting to receive a higher mark or develop a better understanding of your studies, EUREKA will support you as you continue to grow as a learner.

Key Features of EUREKA Class 10 Courses:
AI-enabled doubt solving 24/7 Interactive and personalized learning experience Story-led concept explanations Board exam-focused question practice Instant assessments and feedback Smart progress reports NCERT and CBSE syllabus coverage Flexible self-paced learning Expert faculty mentorship

Explore Now

9.0Supporting Study Materials

This study material CBSE Notes and NCERT Solutions for the Chapter "Surface Areas and Volumes" on Area of Sphere Topics is designed according to the latest CBSE Class 10 Mathematics syllabus and NCERT guidelines. It provides clear explanations of key concepts, definitions, formulas, and important questions to help students understand the total surface area of a sphere, curved and total surface areas of a hemisphere, and prepare effectively for examinations.

CBSE Class 10 Maths Notes Chapter 12 Surface Areas and Volumes	NCERT Solutions for Class 10 Maths Chapter 12: Surface Areas and Volumes

10.0Previous Year Questions on Area of Sphere

Question 1 (CBSE Board): A solid sphere has a radius of 7 cm. Find its total surface area.

Solution: Given, Radius (r) = 7 cm

Total Surface Area of Sphere = 4πr²

= 4 × (22/7) × 7 × 7 = 616 cm²

Answer: Total Surface Area = 616 cm²

11.030-Second Quick Revision: Area of Sphere

Sphere is perfectly round
Radius = r, Surface Area = 4πr²
No edges or vertices
All points are equidistant from centre
Surface area measures outer covering
Used in balls and planets
Remember: Surface Area = 4πr²

12.0Recommended Next Topics

Quadratic Formula

Section Formula

Cone

Surface Area of a Hemisphere

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Surface Area of a Sphere

1.0Master Spherical Geometry in Minutes

Learn how to calculate both the outer surface space and the total internal capacity of a perfectly round three-dimensional object. Understand the fundamental formulas, see how spheres relate to cylinders, and master high-yield board exam questions.

Class: 10 Mathematics (CBSE)
Chapter: Surface Areas and Volumes
Estimated Learning Time: 20–25 Minutes

2.0Learning Outcomes

After completing this lesson, you will be able to:

Define a sphere and understand its unique single-variable property.
State and apply the formula for the Surface Area of a complete sphere.
State and apply the formula for the Volume of a complete sphere.
Differentiate between two-dimensional area measurements and three-dimensional capacity measurements.
Solve board exam problems involving the melting, recasting, and combination of spherical solids.

3.0Area of Sphere

From the smallest cricket ball to the largest celestial body, spheres are all around us and understanding the surfaces related to the sphere is important for solving many real-life applications. The area of a sphere is one such application. The total surface area of a sphere is the area covered by the sphere in a three-dimensional setting.

4.0Understanding the Sphere

The sphere is nothing but a three-dimensional circle in which all the points on the surface lie at equal distances from the centre. In other words, it is the group of certain points at the same distance from the centre. The distance, like a circle, from the centre to the outer surface is known as the radius of the sphere, and the line joining two opposite sides of the sphere passing through the centre is known as the diameter of the sphere.

Diameter= $2 \times R a d i u s$

5.0What is the Formula for Sphere Area?

A sphere is the only three-dimensional figure having the same or equal curved and total surface area. The curved surface area of any 3-D figure is the area of the curved part only, while the total surface area includes the curved surface area along with the area of the base and/or the top part of the figure.

$Surface Area of the sphere(A) = 4 (π) r^{2}$

“The unit for the surface area of the sphere is the same as the square of the sphere's radius while solving the question.”

Area of a Sphere in Diameter

If, while solving any question, the diameter (d) of the sphere is given instead of the radius (r) of the sphere, then the formula for the area of the sphere can be expressed by using the relationship, $r = \frac{d}{2}$ , as:

$Surface Area of the sphere(A) = 4 π (\frac{d}{2})^{2}$

$A = 4 π \times \frac{d ^{2}}{4} = π d^{2}$

Steps to Use the Formula

Step 1: Determine the radius of the sphere and convert the unit into the required unit if needed.

Step 2: Solve the question using the correct values of each component. Always use $π = \frac{22}{7}$ if not mentioned otherwise.

6.0The Surface Area of the Sphere Proof

We can find the area of a sphere derivation by comparing the sphere with the lateral or curved surface area of a cylinder, which is also a 3-D figure. The derivation for the formula is:

Step 1: Comparing the sphere to a cylinder

Consider a sphere fitting perfectly within a cylinder with the same radius(r) as the cylinder and height (h) of the cylinder with the same as the diameter of the sphere such that h = 2r

Step 2: Lateral surface area of a cylinder

Lateral surface area of cylinder = $2 (π) r h$

We know h = 2r

So, Lateral surface area of cylinder = $2 (π) r (2 r)$

The new lateral surface area of a cylinder = $4 (π) r^{2}$

Since the surface area of the sphere = the lateral surface area of the cylinder

Hence,

The surface area of the sphere = $4 (π) r^{2}$

7.0Solved Examples

Problem 1: The surface area of a spherical tank is 1540 cm². Find the radius of the tank.

Solution: It is given that A = 1540cm²

$Surface Area of the sphere(A) = 4 (π) r^{2}$

$1540 = 4 \times \frac{22}{7} \times r^{2}$

$r^{2} = \frac{7 \times 1540}{22 \times 4}$

$r = \frac{7 \times 70}{4} = 3.5 10 c m^{2}$

Problem 2: The outer radius of a hollow spherical shell is 14 cm, and the inner radius is 7 cm. Calculate the surface area of the outer surface and the inner surface of the shell.

Solution: Given that,

Outer radius (R) = 14cm

Inner radius (r) = 7cm

The surface area of the outer surface = $4 π R^{2} = 4 (\frac{22}{7}) 1 4^{2}$

$= 4 \times 22 \times 28 = 2464 c m^{2}$

The surface area of the inner surface $4 π r^{2} = 4 (\frac{22}{7}) 7^{2}$

$= 4 \times 22 \times 7 = 616 c m^{2}$

Problem 3: The volume of a sphere is given as 904.32 cm³. Find the surface area of the sphere.

Solution: We know the Volume of the sphere (V) = $\frac{4}{3} π r^{3}$

$\frac{4}{3} π r^{3} = 904.32$

$r^{3} = \frac{904.32 \times 3}{4 π} \approx 216 c m^{3}$

$r = 3216 = 6 c m$

Now, using Surface Area of the sphere(A) = $4 π r^{2}$

$A = 4 \times \frac{22}{7} \times 6^{2} = \frac{3168}{7}$

$A = 452.38 c m^{2}$

8.0EUREKA by ALLEN – Learn Better, Score Highe

EUREKA by ALLEN is designed to simplify, enrich, and enhance your experience in Class 10. Through the use of fun and engaging video lessons, regular practice tests, and immediate help for any doubts you may have regarding the material; students have a firm understanding of the concepts they are studying and feel confident in their preparation for their board exams. No matter if you are attempting to receive a higher mark or develop a better understanding of your studies, EUREKA will support you as you continue to grow as a learner.

Key Features of EUREKA Class 10 Courses:
AI-enabled doubt solving 24/7 Interactive and personalized learning experience Story-led concept explanations Board exam-focused question practice Instant assessments and feedback Smart progress reports NCERT and CBSE syllabus coverage Flexible self-paced learning Expert faculty mentorship

Explore Now

9.0Supporting Study Materials

This study material CBSE Notes and NCERT Solutions for the Chapter "Surface Areas and Volumes" on Area of Sphere Topics is designed according to the latest CBSE Class 10 Mathematics syllabus and NCERT guidelines. It provides clear explanations of key concepts, definitions, formulas, and important questions to help students understand the total surface area of a sphere, curved and total surface areas of a hemisphere, and prepare effectively for examinations.

CBSE Class 10 Maths Notes Chapter 12 Surface Areas and Volumes	NCERT Solutions for Class 10 Maths Chapter 12: Surface Areas and Volumes

10.0Previous Year Questions on Area of Sphere

Question 1 (CBSE Board): A solid sphere has a radius of 7 cm. Find its total surface area.

Solution: Given, Radius (r) = 7 cm

Total Surface Area of Sphere = 4πr²

= 4 × (22/7) × 7 × 7 = 616 cm²

Answer: Total Surface Area = 616 cm²

11.030-Second Quick Revision: Area of Sphere

Sphere is perfectly round
Radius = r, Surface Area = 4πr²
No edges or vertices
All points are equidistant from centre
Surface area measures outer covering
Used in balls and planets
Remember: Surface Area = 4πr²

12.0Recommended Next Topics

Quadratic Formula

Section Formula

Cone

Surface Area of a Hemisphere

Frequently Asked Questions

What is the unit for the surface area of a sphere?

Is the surface area of a sphere dependent on its volume?

What happens to the surface area of a sphere if its volume is halved?

Can the surface area of a sphere be negative in any case?

Frequently Asked Questions

What is the unit for the surface area of a sphere?

Is the surface area of a sphere dependent on its volume?

What happens to the surface area of a sphere if its volume is halved?

Can the surface area of a sphere be negative in any case?

Join ALLEN!

Surface Area of a Sphere

1.0Master Spherical Geometry in Minutes

2.0Learning Outcomes

3.0Area of Sphere

4.0Understanding the Sphere

5.0What is the Formula for Sphere Area?

Area of a Sphere in Diameter

6.0The Surface Area of the Sphere Proof

7.0Solved Examples

8.0EUREKA by ALLEN – Learn Better, Score Highe

9.0Supporting Study Materials

10.0Previous Year Questions on Area of Sphere

11.030-Second Quick Revision: Area of Sphere

12.0Recommended Next Topics

Table of Contents

Join ALLEN!

Surface Area of a Sphere

1.0Master Spherical Geometry in Minutes

2.0Learning Outcomes

3.0Area of Sphere

4.0Understanding the Sphere

5.0What is the Formula for Sphere Area?

Area of a Sphere in Diameter

6.0The Surface Area of the Sphere Proof

7.0Solved Examples

8.0EUREKA by ALLEN – Learn Better, Score Highe

9.0Supporting Study Materials

10.0Previous Year Questions on Area of Sphere

11.030-Second Quick Revision: Area of Sphere

12.0Recommended Next Topics

Table of Contents