22. Hemisphere Formulas
A hemisphere is half of a sphere. It is formed when a sphere is divided into two equal parts through its centre.
A hemisphere has:
- One curved surface
- One circular flat base
Let r represent the radius of the hemisphere.
The provided Mensuration Formula PDF gives the following hemisphere formulas for volume, curved surface area and total surface area. :contentReference[oaicite:1]{index=1}
1. Hemisphere Diagram
2. What Is a Hemisphere?
The word hemisphere means half of a sphere.
If a sphere is divided into two equal parts through its centre, each part is called a hemisphere.
Important: A hemisphere has both a curved surface and a circular base.
3. Hemisphere Formulas
Volume of Hemisphere
The volume of a hemisphere is half the volume of a complete sphere.
Curved Surface Area of Hemisphere
This formula gives only the curved part of the hemisphere. The circular base is not included.
Total Surface Area of Hemisphere
The total surface area includes:
- Curved surface area
- Circular base area
4. Example 1 — Volume of a Hemisphere
Find the volume of a hemisphere whose radius is 7 cm. Take π = 22/7.
Step 1: Write the formula
Step 2: Substitute the values
Step 3: Calculate
Answer: 718.67 cm³ approximately.
5. Example 2 — Curved Surface Area
Find the curved surface area of a hemisphere with radius 7 cm. Take π = 22/7.
Answer: 308 cm².
6. Example 3 — Total Surface Area
Find the total surface area of a hemisphere whose radius is 7 cm. Take π = 22/7.
Answer: 462 cm².
7. CSA vs TSA of Hemisphere
| Type | Formula | What Is Included? |
|---|---|---|
| CSA | 2πr² | Only curved surface |
| TSA | 3πr² | Curved surface + circular base |
8. Why Is TSA = 3πr²?
The hemisphere has two types of surfaces:
- Curved Surface Area
- Circular Base Area
Therefore:
9. Sphere vs Hemisphere
| Sphere | Hemisphere |
|---|---|
| Complete spherical solid | Half of a sphere |
| Volume = ⁴⁄₃πr³ | Volume = ²⁄₃πr³ |
| Surface Area = 4πr² | CSA = 2πr² |
| No flat circular base | Has one circular base |
| Surface area = 4πr² | TSA = 3πr² |
10. Symbols Used
| Symbol | Meaning |
|---|---|
| r | Radius of hemisphere |
| π | Pi, generally taken as 22/7 or 3.14 |
| CSA | Curved Surface Area |
| TSA | Total Surface Area |
11. Radius and Diameter
If the diameter of the hemisphere is given instead of the radius, first find the radius.
where d is the diameter.
12. Example 4 — When Diameter Is Given
The diameter of a hemisphere is 14 cm. Find its curved surface area. Take π = 22/7.
Step 1: Find radius
Step 2: Apply CSA formula
Answer: 308 cm².
13. Practice Questions
1. Find the volume of a hemisphere with radius 7 cm. Take π = 22/7.
2. Find the CSA of a hemisphere with radius 14 cm. Take π = 22/7.
3. Find the TSA of a hemisphere with radius 14 cm. Take π = 22/7.
4. The diameter of a hemisphere is 28 cm. Find its volume. Take π = 22/7.
5. The radius of a hemisphere is 21 cm. Find its CSA. Take π = 22/7.
6. The radius of a hemisphere is 7 cm. Find its TSA. Take π = 22/7.
14. Answers
| Question | Answer |
|---|---|
| 1 | 718.67 cm³ approximately |
| 2 | 1232 cm² |
| 3 | 1848 cm² |
| 4 | 5749.33 cm³ approximately |
| 5 | 2772 cm² |
| 6 | 462 cm² |
15. Important Exam Shortcuts
Shortcut 1: Hemisphere is half of a sphere.
Shortcut 2: Curved surface area does not include the circular base.
Shortcut 3: Total surface area includes the circular base.
Shortcut 4: If diameter is given, first convert it into radius.
16. Common Mistakes to Avoid
1. Do not use the sphere's surface-area formula 4πr² directly for the TSA of a hemisphere.
2. Remember that hemisphere TSA includes the flat circular base.
3. Do not confuse CSA and TSA.
4. Always check whether the question gives radius or diameter.
5. Volume is measured in cubic units such as cm³ or m³.
6. Surface area is measured in square units such as cm² or m².
17. One-Minute Revision
Source: Complete Mensuration Formulas PDF, final page — “SURFACE AREA & VOLUME”. The PDF specifically lists Hemisphere with Volume = ²⁄₃πr³, Curved Surface Area = 2πr² and Total Surface Area = 3πr². :contentReference[oaicite:2]{index=2}