🔵 Sphere and Hemisphere
(With Practical Examples, Formulas & Comparison – Beginner Friendly)
Sphere
✅ Definition:
A sphere is a perfectly round 3D shape in which all points on the surface are at equal distance from the center.
👉 It has:
No edges
No verticesOnly one curved surface
🟢 Practical Examples:
Football
Globe (Earth model)
Orange
Marble
Soap bubble
📐 Important Formulas (Sphere)
Let radius = r
Surface Area = 4πr²
Volume = (4/3)πr³
Hemisphere
✅ Definition:
A hemisphere is half of a sphere.
👉 If you cut a sphere into two equal parts, each part is a hemisphere.
It has:
One curved surface
One circular flat base🟢 Practical Examples:
Bowl
Dome of a building
Half-cut watermelon
Igloo house
📐 Important Formulas (Hemisphere)
Let radius = r
Curved Surface Area (CSA) = 2πr²
Total Surface Area (TSA) = 3πr²
Volume = (2/3)πr³
👉 TSA = Curved area + Base area
= 2πr² + πr²
= 3πr²
Comparison Table
| Feature | Sphere | Hemisphere |
|---|---|---|
| Shape | Fully round | Half round |
| Flat surface | ❌ None | ✅ One circular base |
| Surface Area | 4πr² | 3πr² (TSA) |
| Volume | (4/3)πr³ | (2/3)πr³ |
| Example | Football | Bowl |
Simple Numerical Example
Example: Radius = 7 cm
(Use π = 22/7)
🔵 Sphere
Surface Area = 4 × 22/7 × 49
= 616 cm²
Volume = (4/3) × 22/7 × 343
= 1437.33 cm³
🟢 Hemisphere
Curved Surface Area = 2 × 22/7 × 49
= 308 cm²
Total Surface Area = 3 × 22/7 × 49
= 462 cm²
Volume = (2/3) × 22/7 × 343
= 718.67 cm³
🔥 Important Exam Points
✔ Volume of hemisphere is half of sphere
✔ Surface area of sphere = 2 × TSA of hemisphere
✔ No edges and vertices in sphere
✔ Hemisphere has 1 flat circular base
One-Line Difference
👉 Sphere = Complete round ball
👉 Hemisphere = Half ball
Why is the Area Formula of a Sphere = 4πr²?
Many students memorize formulas but don’t know why they are like that. Let’s understand in an easy way 👇
First Recall: Area of a Circle
Area of a circle = πr²
This is basic and you already know it.
Now think:
A sphere is made of curved surface all around — like many small circles wrapped together.
Big Idea Behind Sphere Surface Area
A sphere’s surface area = 4 times the area of its biggest circle (great circle).
Great circle area = πr²
So,
Surface Area of Sphere = 4 × πr²
= 4πr²
Simple Practical Understanding
Imagine:
Take 4 circular papers of radius r.
Their total area = 4πr².That total area equals the entire outer covering of a ball (sphere).
👉 That is why sphere area = 4πr².
Why Hemisphere Area = 3πr² ?
Hemisphere = Half sphere
Full sphere area = 4πr²
Half curved area = 2πr²
But hemisphere also has a flat circular base.
Base area = πr²
So total surface area:
2πr² + πr²
= 3πr²
Why Curved Surface Area of Hemisphere = 2πr² ?
Because it is exactly half of sphere’s curved surface.
Sphere curved surface = 4πr²
Half of that = 2πr²
Why These Formulas Look Similar to Cylinder?
Cylinder curved surface area = 2πrh
If height of cylinder = 2r (diameter),
CSA becomes 2πr(2r) = 4πr²
Which is same as sphere surface area!
This is not coincidence — sphere and cylinder are mathematically connected (proved by ancient mathematicians).
Short Concept Summary
Circle area → πr²
Sphere surface → 4 circles → 4πr²
Hemisphere curved → 2πr²
Hemisphere total → 3πr²
Final Understanding
👉 Sphere area is 4πr² because its curved surface equals four times the area of its great circle.