Sphere and Hemisphere – Surface Area and Volume

Geometry questions on Sphere and Hemisphere are common in competitive examinations. The key is to understand the difference between surface area and volume, and especially the difference between a sphere and a hemisphere.

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1. What is a Sphere?

A sphere is a perfectly round three-dimensional solid in which every point on its surface is at the same distance from its centre.

That distance is called the radius (r).

Examples: ball, globe, marble, etc.

Important Terms

Term	Meaning
Radius (r)	Distance from centre to surface
Diameter (d)	Distance across the sphere through centre
Relation	d = 2r

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2. Surface Area of a Sphere

A sphere has only one continuous curved surface.

Formula:

$$\boxed{\text{Surface Area of Sphere}=4\pi r^2}$$

Since $d=2r$, we can also write:

$$\boxed{\text{Surface Area}=\pi d^2}$$

Example

Find the surface area of a sphere having radius 7 cm.

$$SA=4\pi r^2$$ $$=4\times\frac{22}{7}\times7\times7$$ $$=616\text{ cm}^2$$

Answer = 616 cm²

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3. Volume of a Sphere

The volume tells us how much space is occupied by the sphere.

Formula:

$$\boxed{\text{Volume of Sphere}=\frac{4}{3}\pi r^3}$$

Example

Find the volume of a sphere with radius 3 cm.

$$V=\frac43\pi r^3$$ $$=\frac43\times\pi\times3^3$$ $$=36\pi$$

Therefore,

$$\boxed{V=36\pi\text{ cm}^3}$$

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4. What is a Hemisphere?

The word hemi means half.

Therefore, a hemisphere is half of a sphere.

Imagine cutting a sphere exactly through its centre. Each half obtained is a hemisphere.

Examples: half of a ball, dome-shaped objects, bowl-like structures, etc.

A hemisphere has:

1. One curved surface

2. One circular flat base

This flat circular base is extremely important when calculating surface area.

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5. Curved Surface Area of a Hemisphere

The curved surface area of a sphere is:

$$4\pi r^2$$

A hemisphere is half of a sphere.

Therefore:

$$\boxed{\text{CSA of Hemisphere}=2\pi r^2}$$

Example

Find the curved surface area of a hemisphere of radius 7 cm.

$$CSA=2\pi r^2$$ $$=2\times\frac{22}{7}\times7\times7$$ $$=308\text{ cm}^2$$

Answer = 308 cm²

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6. Total Surface Area of a Hemisphere

This is where students frequently make mistakes.

A hemisphere has:

• Curved surface area = $2\pi r^2$

• Circular base area = $\pi r^2$

Therefore:

$$TSA=2\pi r^2+\pi r^2$$ $$\boxed{\text{TSA of Hemisphere}=3\pi r^2}$$

Example

Find the total surface area of a hemisphere of radius 7 cm.

$$TSA=3\pi r^2$$ $$=3\times\frac{22}{7}\times7\times7$$ $$=462\text{ cm}^2$$

Answer = 462 cm²

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7. Volume of a Hemisphere

Since a hemisphere is exactly half of a sphere:

$$V=\frac12\times\frac43\pi r^3$$

Therefore:

$$\boxed{\text{Volume of Hemisphere}=\frac23\pi r^3}$$

Example

Find the volume of a hemisphere with radius 3 cm.

$$V=\frac23\pi r^3$$ $$=\frac23\pi(3)^3$$ $$=18\pi$$

Therefore:

$$\boxed{18\pi\text{ cm}^3}$$

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8. Sphere vs Hemisphere – Comparison

Property	Sphere	Hemisphere
Shape	Complete round solid	Half of a sphere
Curved surface area	$4\pi r^2$	$2\pi r^2$
Flat circular base	No	Yes
Total surface area	$4\pi r^2$	$3\pi r^2$
Volume	$\frac{\mathrm{4/}}{3}\pi r^3$	$\frac{\mathrm{2/}}{3}\pi r^3$
Diameter	$2r$	Circular base diameter = $2r$

Easy Memory Trick

Sphere:

$$SA=4\pi r^{\mathrm{2 }}$$

Hemisphere:

$$CSA=2\pi r^2$$

Remember:

Hemisphere = half sphere, but TSA is NOT half of sphere's surface area because the flat circular base must be added.

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9. Important Difference: CSA and TSA

For a hemisphere:

Curved Surface Area

Only the curved portion is considered:

$$\boxed{{\displaystyle CSA=2\pi {\mathrm{r2}}^{}}}$$

Total Surface Area

Curved portion + circular base:

$$\boxed{{\displaystyle TSA=3\pi r^2}}$$

This distinction is frequently tested in competitive examinations.

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10. Important Formula Summary

Quantity	Formula
Diameter of sphere	$d=2r$
Sphere Surface Area	$4\pi r^2$
Sphere Volume	$\frac43\pi r^3$
Hemisphere CSA	$2\pi r^2$
Hemisphere TSA	$3\pi r^2$
Hemisphere Volume	$\frac23\pi r^3$
Circle/base area	$\pi r^2$

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11. Important Exam Tips

Tip 1: Check what the question asks

If the question says curved surface area, don't include the circular base.

If it says total surface area, include the base.

Tip 2: Radius and diameter

If diameter is given:

$$r=\frac d2$$

Never substitute diameter directly in a radius formula.

Tip 3: Area vs Volume

Surface area uses:

$$r^2$$

Volume uses:

$$r^3$$

So:

• Area → square units such as cm²

• Volume → cubic units such as cm³

Tip 4: Use $\pi=\frac{22}{7}$ when appropriate

If the radius or diameter is a multiple of 7, $\frac{22}{7}$ usually makes calculation easier.

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12. MCQs – Sphere and Hemisphere

Q1. The surface area of a sphere of radius 7 cm is:

A) 154 cm²
B) 308 cm²
C) 462 cm²
D) 616 cm²

Answer: D) 616 cm²

Q2. The volume of a sphere of radius 3 cm is:

A) $9\pi$ cm³
B) $18\pi$ cm³
C) $36\pi$ cm³
D) $27\pi$ cm³

Answer: C) $36\pi$ cm³

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Q3. The curved surface area of a hemisphere of radius 7 cm is:

A) 154 cm²
B) 308 cm²
C) 462 cm²
D) 616 cm²

Answer: B) 308 cm²

Q4. The total surface area of a hemisphere of radius 7 cm is:

A) 154 cm²
B) 308 cm²
C) 462 cm²
D) 616 cm²

Answer: C) 462 cm²

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Q5. The volume of a hemisphere of radius $r$ is:

A) $\frac{4}{3}\pi r^3$
B) $2\pi r^3$
C) $\frac{2}{3}\pi r^3$
D) $3\pi r^3$

Answer: C) $\frac{2}{3}\pi r^3$

Q6. The total surface area of a hemisphere of radius $r$ is:

A) $2\pi r^2$
B) $3\pi r^2$
C) $4\pi r^2$
D) $\pi r^2$

Answer: B) $3\pi r^2$

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Q7. A sphere has radius 5 cm. Its diameter is:

A) 2.5 cm
B) 5 cm
C) 10 cm
D) 25 cm

Answer: C) 10 cm

Q8. The surface area of a sphere is $4\pi r^2$. If its radius is doubled, the surface area becomes:

A) 2 times
B) 3 times
C) 4 times
D) 8 times

Answer: C) 4 times

Explanation: Surface area depends on $r^2$.

$$(2r)^2=4r^2$$

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Q9. If the radius of a sphere is doubled, its volume becomes:

A) 2 times
B) 4 times
C) 6 times
D) 8 times

Answer: D) 8 times

Q10. A hemisphere has:

A) Only a curved surface
B) Only a flat surface
C) A curved surface and a circular base
D) Two circular bases

Answer: C) A curved surface and a circular base

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13. Statement-Type MCQs

Q11. Consider the following statements:

1. The surface area of a sphere is $4\pi r^2$.

2. The volume of a sphere is $\frac43\pi r^3$.

3. The curved surface area of a hemisphere is $3\pi r^2$.

4. The total surface area of a hemisphere is $3\pi r^2$.

Which statements are correct?

A) 1 and 2 only
B) 1, 2 and 4 only
C) 2 and 3 only
D) All are correct

Answer: B) 1, 2 and 4 only

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Q12. Consider the following statements regarding a hemisphere:

1. It is half of a sphere.

2. It has a circular flat base.

3. Its curved surface area is $2\pi r^2$.

4. Its total surface area is $3\pi r^2$.

Which of the above are correct?

A) 1 and 2 only
B) 2 and 3 only
C) 1, 2 and 3 only
D) 1, 2, 3 and 4

Answer: D) 1, 2, 3 and 4

One-line memory trick:

Sphere SA → 4πr²

Sphere Volume → 4/3πr³
Hemisphere CSA → 2πr²
Hemisphere TSA → 3πr²
Hemisphere Volume → 2/3πr³