Surface Area of Sphere

Class 9Surface Areas and Volumes

A sphere is a three-dimensional solid where every point on the surface is at the same distance from the centre. The surface area of a sphere is the total area of the curved surface that encloses it.

Unlike a cube or cuboid, a sphere has only one curved surface with no edges or vertices. Its surface area depends solely on the radius.

The formula for the surface area of a sphere is 4πr², which equals exactly four times the area of a great circle. This topic is covered in NCERT Class 9 Mathematics, Chapter: Surface Areas and Volumes.

What is Surface Area of Sphere?

Definition: The surface area of a sphere is the total area of the curved surface enclosing the sphere.

For a sphere of radius r:

Surface Area = 4πr²

Where:

r = radius of the sphere
π = 22/7 or 3.14159...
The result is in square units (cm², m², etc.)

Important:

A sphere has NO flat surface — the entire area is curved.
CSA = TSA = 4πr² (there is no separate lateral or flat surface).
Surface area of a sphere = 4 × area of a great circle (a great circle has area πr²).
Doubling the radius increases the surface area 4 times (since it depends on r²).

Surface Area of Sphere Formula

Key Formulas:

1. Surface area of a sphere:

SA = 4πr²

2. Surface area in terms of diameter:

Since r = d/2: SA = 4π(d/2)² = πd²

3. Curved Surface Area of hemisphere:

CSA of hemisphere = 2πr²

4. Total Surface Area of hemisphere:

TSA of hemisphere = 3πr²

5. Finding radius from surface area:

r² = SA / (4π)
r = √[SA / (4π)]

Derivation and Proof

Derivation of Surface Area of Sphere (Experimental Method):

Step 1: Take a sphere and wind thread

Take a sphere of radius r and a piece of thread.
Wind the thread tightly around the surface of the sphere, covering it completely.

Step 2: Measure the thread

Unwind the thread and measure its total length.
This length represents the surface area when wound into a strip of width equal to the thread’s thickness.

Step 3: Compare with circles

Draw a circle of the same radius r on paper (this is a great circle).
Wind thread over this circle to fill it completely.

Step 4: Count the circles

The thread that covered the sphere fills exactly 4 such circles.
Area of one great circle = πr²
Surface area of sphere = 4 × πr² = 4πr²

Step 5: Hemisphere surface area

Curved surface area of hemisphere = (1/2) × 4πr² = 2πr²
The flat circular face has area = πr²
Total surface area of hemisphere = 2πr² + πr² = 3πr²

Types and Properties

Types of surface area problems involving spheres:

1. Full Sphere

SA = 4πr²
CSA = TSA (no flat surface)

2. Hemisphere — Curved Surface Area

CSA = 2πr² (only the dome part)
Used when painting or coating just the curved surface.

3. Hemisphere — Total Surface Area

TSA = 3πr² (dome + flat circular base)
Used when the entire outer surface needs covering.

4. Hollow Sphere

Outer surface area = 4πR²
Inner surface area = 4πr²
Total area = 4πR² + 4πr² (if both surfaces are exposed)

5. Cost-based problems

Find SA, then multiply by cost per unit area.
Common: painting, polishing, wrapping, coating.

Solved Examples

Example 1: Example 1: Surface area given radius

Problem: Find the surface area of a sphere of radius 7 cm. (Use π = 22/7)

Solution:

Given:

r = 7 cm

Using SA = 4πr²:

SA = 4 × (22/7) × 7²
SA = 4 × (22/7) × 49
SA = 4 × 22 × 7
SA = 616 cm²

Answer: Surface area = 616 cm².

Example 2: Example 2: Surface area given diameter

Problem: Find the surface area of a sphere of diameter 21 cm. (Use π = 22/7)

Solution:

Given:

d = 21 cm, so r = 10.5 cm

Using SA = 4πr²:

SA = 4 × (22/7) × (10.5)²
SA = 4 × (22/7) × 110.25
SA = 4 × 346.5
SA = 1386 cm²

Answer: Surface area = 1386 cm².

Example 3: Example 3: Finding radius from surface area

Problem: The surface area of a sphere is 2464 cm². Find the radius. (Use π = 22/7)

Solution:

Given:

SA = 2464 cm²

Using SA = 4πr²:

2464 = 4 × (22/7) × r²
2464 = (88/7) × r²
r² = 2464 × 7 / 88 = 17248/88 = 196
r = √196 = 14 cm

Answer: Radius = 14 cm.

Example 4: Example 4: CSA of hemisphere

Problem: Find the curved surface area of a hemisphere of radius 14 cm. (Use π = 22/7)

Solution:

Given:

r = 14 cm

Using CSA = 2πr²:

CSA = 2 × (22/7) × 14²
CSA = 2 × (22/7) × 196
CSA = 2 × 22 × 28
CSA = 1232 cm²

Answer: Curved surface area = 1232 cm².

Example 5: Example 5: TSA of hemisphere

Problem: Find the total surface area of a solid hemisphere of radius 3.5 cm. (Use π = 22/7)

Solution:

Given:

r = 3.5 cm

Using TSA = 3πr²:

TSA = 3 × (22/7) × (3.5)²
TSA = 3 × (22/7) × 12.25
TSA = 3 × 38.5
TSA = 115.5 cm²

Answer: Total surface area = 115.5 cm².

Example 6: Example 6: Cost of painting a sphere

Problem: A spherical ball has a radius of 10.5 cm. Find the cost of painting it at Rs 4 per cm². (Use π = 22/7)

Solution:

Given:

r = 10.5 cm, cost = Rs 4 per cm²

Surface area:

SA = 4 × (22/7) × (10.5)²
SA = 4 × (22/7) × 110.25
SA = 4 × 346.5 = 1386 cm²

Cost:

Cost = 1386 × 4 = Rs 5,544

Answer: Cost of painting = Rs 5,544.

Example 7: Example 7: Ratio of surface areas

Problem: The radii of two spheres are in the ratio 2 : 3. Find the ratio of their surface areas.

Solution:

Given:

r₁ : r₂ = 2 : 3

Ratio of surface areas:

SA₁ : SA₂ = 4πr₁² : 4πr₂²
= r₁² : r₂²
= 2² : 3²
= 4 : 9

Answer: Ratio of surface areas = 4 : 9.

Example 8: Example 8: Sphere from cylinder

Problem: A cylinder of radius 7 cm and height 28 cm is melted to form a sphere. Find the radius and surface area of the sphere. (Use π = 22/7)

Solution:

Volume of cylinder:

V = πr²h = (22/7) × 49 × 28 = 4312 cm³

Finding sphere radius:

(4/3)πR³ = 4312
(4/3) × (22/7) × R³ = 4312
R³ = 4312 × 21 / 88 = 1029
R = ³√1029 ≈ 10.1 cm

Surface area:

SA = 4πR² = 4 × (22/7) × 102.01 ≈ 1282.4 cm²

Answer: Radius ≈ 10.1 cm; Surface area ≈ 1282.4 cm².

Example 9: Example 9: Comparing sphere and cube surface areas

Problem: A sphere of radius 3.5 cm fits exactly inside a cube. Find the ratio of the surface area of the sphere to that of the cube.

Solution:

Given:

r = 3.5 cm ⇒ side of cube = diameter = 7 cm

Surface area of sphere:

SA_sphere = 4πr² = 4 × (22/7) × 12.25 = 154 cm²

Surface area of cube:

SA_cube = 6a² = 6 × 49 = 294 cm²

Ratio:

154 : 294 = 11 : 21

Answer: Ratio = 11 : 21.

Example 10: Example 10: Doubling the radius

Problem: If the radius of a sphere is doubled, by what factor does the surface area increase?

Solution:

Original surface area:

SA = 4πr²

New surface area (radius = 2r):

SA' = 4π(2r)² = 4π(4r²) = 16πr²
SA' = 4 × 4πr² = 4 × SA

Answer: The surface area increases by a factor of 4.

Real-World Applications

Applications of Surface Area of Sphere:

Manufacturing: Calculating the amount of material (rubber, metal, leather) needed to make balls, balloons, and globes.
Painting and coating: Determining the amount of paint needed to cover domes, spherical tanks, and decorative spheres.
Architecture: Estimating roofing material for domed buildings, planetariums, and observatories.
Medicine: Calculating dosage for spherical pills and drug delivery microspheres based on surface-to-volume ratio.
Astronomy: Estimating the surface area of planets and stars for radiation calculations.
Packaging: Designing spherical containers where surface area determines material cost.

Key Points to Remember

Surface area of sphere = 4πr², where r is the radius.
Surface area of sphere = 4 × area of a great circle.
A sphere has only a curved surface — CSA = TSA = 4πr².
CSA of hemisphere = 2πr² (curved part only).
TSA of hemisphere = 3πr² (curved + flat circular base).
In terms of diameter: SA = πd².
Doubling the radius increases surface area 4 times (depends on r²).
Surface area is always in square units (cm², m²).
For cost problems: Total cost = Surface area × rate per unit area.
The formula was experimentally derived by comparing the thread covering a sphere with that covering great circles.

Practice Problems

Find the surface area of a sphere of radius 3.5 cm. (Use π = 22/7)
The surface area of a sphere is 5544 cm². Find the radius. (Use π = 22/7)
Find the TSA of a solid hemisphere of radius 21 cm.
A hemispherical dome has a radius of 7 m. Find the cost of whitewashing its curved surface at Rs 3 per m².
The radii of two spheres are 3 cm and 5 cm. Find the ratio of their surface areas.
A sphere of radius 5 cm is melted and drawn into a wire of radius 0.1 cm. Find the length of the wire.
Find the diameter of a sphere whose surface area is 1386 cm².
A hollow sphere has outer radius 12 cm and inner radius 10 cm. Find the total surface area exposed (inner + outer + rim).

Frequently Asked Questions

Q1. What is the formula for the surface area of a sphere?

Surface area of a sphere = 4πr², where r is the radius. In terms of diameter, it is πd².

Q2. What is the difference between CSA and TSA of a sphere?

For a full sphere, CSA = TSA = 4πr² because it has no flat surface. For a hemisphere, CSA = 2πr² (curved only) and TSA = 3πr² (curved + flat base).

Q3. Why is the surface area of a sphere 4 times the area of a great circle?

This is established experimentally: the thread needed to cover a sphere exactly fills 4 great circles of the same radius. Mathematically, integrating the surface elements of a sphere gives 4πr² = 4 × πr².

Q4. How does the surface area change if the radius is tripled?

The surface area becomes 9 times the original. Since SA depends on r², the new SA = 4π(3r)² = 9 × 4πr².

Q5. How do you find the radius from the surface area?

Use r² = SA/(4π), then take the square root. For example, if SA = 616 cm² and π = 22/7: r² = 616 × 7/88 = 49, so r = 7 cm.

Q6. What is the surface area of the Earth?

The Earth’s radius is approximately 6,371 km. Surface area = 4π(6371)² ≈ 510 million km². This is a direct application of the formula.

Q7. Is surface area of sphere in the CBSE Class 9 syllabus?

Yes. Surface area of sphere and hemisphere are covered in CBSE Class 9 Mathematics, Chapter: Surface Areas and Volumes.

Q8. What is a great circle?

A great circle is the largest circle that can be drawn on a sphere. It passes through the centre of the sphere and divides it into two equal hemispheres. The equator is a great circle of the Earth.

Surface Area of Sphere

What is Surface Area of Sphere?

Surface Area of Sphere Formula

Derivation and Proof

Types and Properties

Solved Examples

Example 1: Example 1: Surface area given radius

Example 2: Example 2: Surface area given diameter

Example 3: Example 3: Finding radius from surface area

Example 4: Example 4: CSA of hemisphere

Example 5: Example 5: TSA of hemisphere

Example 6: Example 6: Cost of painting a sphere

Example 7: Example 7: Ratio of surface areas

Example 8: Example 8: Sphere from cylinder

Example 9: Example 9: Comparing sphere and cube surface areas

Example 10: Example 10: Doubling the radius

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. What is the formula for the surface area of a sphere?

Q2. What is the difference between CSA and TSA of a sphere?

Q3. Why is the surface area of a sphere 4 times the area of a great circle?

Q4. How does the surface area change if the radius is tripled?

Q5. How do you find the radius from the surface area?

Q6. What is the surface area of the Earth?

Q7. Is surface area of sphere in the CBSE Class 9 syllabus?

Q8. What is a great circle?

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