How to Find the Curved and Total Surface Area of a Hemisphere?
The concept of Area of Hemisphere plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you’re calculating the coating area of a bowl or solving 3D geometry questions in board exams and Olympiads, a clear understanding of the area of hemisphere is essential.
What Is Area of Hemisphere?
A hemisphere is exactly half of a sphere. Imagine cutting a solid ball into two equal parts; each part is a hemisphere. Just like a sphere, its size and area are calculated based on its radius. In Maths, the area of hemisphere tells us how much surface the outer side (curved part) or the whole hemisphere (curved + flat base) covers. You’ll find this concept applied while measuring hemispherical domes, solving mensuration questions, and in calculating the exposed area of vessels or physical models.
Key Formula for Area of Hemisphere
Here’s the standard formula:
Curved Surface Area (CSA): \( 2\pi r^2 \)
Total Surface Area (TSA): \( 3\pi r^2 \)
Where r is the radius of the hemisphere.
Explain: CSA finds the area of the dome part, and TSA includes both the dome and the circular base.
Types of Surface Area in Hemispheres
| Type | Formula | When To Use |
|---|---|---|
| Curved Surface Area (CSA) | 2πr² | Only the dome (no base), e.g. coconut shell, dish, cap |
| Total Surface Area (TSA) | 3πr² | Full hemisphere (dome + flat base) |
| Base Area | πr² | Just the flat circular base |
Step-by-Step Illustration
- Suppose the radius of a solid hemisphere is 7 cm. Find its total surface area (TSA). Use π = 22/7.
2. Insert values: TSA = 3 × (22/7) × 7 × 7
3. Simplify: TSA = 3 × 22 × 7
4. Multiply: 3 × 22 = 66; 66 × 7 = 462
5. So, TSA = 462 cm²
Cross-Disciplinary Usage
Area of hemisphere is not only useful in Maths but also plays an important role in Physics (surface area for thermal calculations), Computer Science (3D graphics), and Engineering (measuring domes or tanks). Students preparing for JEE, NEET, and various Olympiads encounter this concept regularly.
Curved vs. Total Surface Area – At a Glance
- CSA refers just to the outer dome, not the flat base.
- TSA means the dome plus the base.
- Tip: Most board exam questions will mention “Find TSA” for the entire hemisphere, but “Find CSA” when the base is open or not counted.
Formula Table for Quick Revision
| Formula Name | Formula |
|---|---|
| Curved Surface Area (CSA) | 2πr² |
| Base Area | πr² |
| Total Surface Area (TSA) | 3πr² |
Solved Examples
Example 1: Find the curved surface area of a hemisphere of radius 5 cm. Take π = 3.14.
Solution:
1. Write the CSA formula: CSA = 2πr²2. Insert the radius: CSA = 2 × 3.14 × 5 × 5
3. Multiply: 2 × 3.14 = 6.28; then 6.28 × 25 = 157
4. Final Answer: 157 cm²
Example 2: A hemispherical bowl has a radius of 10 cm. Find its total surface area.
1. TSA formula: TSA = 3πr²2. Substitute: TSA = 3 × 3.14 × 10 × 10 = 3 × 3.14 × 100 = 942
3. Final Answer: 942 cm²
Speed Trick or Vedic Shortcut
To quickly check if you must add the base, remember: If the hemisphere is “closed” at the base, use TSA. If it's “open,” use CSA only. Always check the question!
Try These Yourself
- Calculate the CSA of a hemisphere with radius 8 cm using π = 3.14.
- If the TSA of a hemisphere is 706.5 cm², find its radius.
- Is the area of the base always included in “TSA”? Why or why not?
- List three objects in daily life shaped like hemispheres.
Frequent Errors and Misunderstandings
- Forgetting to add the base area for TSA.
- Using wrong value for π (consistency is key).
- Mixing up TSA and CSA; check the “open or closed” base in the question.
- Leaving out square units (should always be cm², m², etc.).
Relation to Other Concepts
The idea of area of hemisphere connects closely with surface area of a sphere and volume of hemisphere. Understanding area also helps you solve problems on menuration, geometry, and even surface painting or design estimation. If you know the area of a circle formula, it’s easy to relate to the base area calculation!
Real-Life Applications
- Measuring paint needed for hemispherical domes and ceilings.
- Estimating material for bowls, caps, and water tanks.
- Calculating energy transfer across hemispherical surfaces in science.
Classroom Tip
A quick way to remember: “TSA means Three Parts Added” (Dome + Flat Base = 3πr²). Vedantu’s teachers use this trick in live classes to help students recall easily during exams and MCQs.
Wrapping It All Up
We explored Area of Hemisphere—from definition, formula, types, solved examples, speed tricks, and common mistakes. Practice these with Vedantu’s hemisphere calculator and seek more explanations through Mensuration topics to become confident in solving any hemisphere question.
Useful Internal Links
FAQs on Area of Hemisphere: Formula, Curved and Total Surface Calculation
1. What is the formula for the curved surface area of a hemisphere?
The curved surface area (CSA) of a hemisphere is given by the formula: CSA = 2πr², where r represents the radius of the hemisphere.
2. What is the formula for the total surface area of a hemisphere?
The total surface area (TSA) of a hemisphere includes both the curved surface and the flat circular base. The formula is: TSA = 3πr², where r is the radius of the hemisphere.
3. What is the difference between the curved surface area and the total surface area of a hemisphere?
The curved surface area only considers the area of the curved portion of the hemisphere, while the total surface area includes both the curved surface and the flat circular base. The TSA is always greater than the CSA.
4. How do I calculate the base area of a hemisphere?
The base of a hemisphere is a circle. Therefore, its area is calculated using the formula for the area of a circle: Area = πr², where r is the radius of the hemisphere.
5. What units are used to express the surface area of a hemisphere?
Surface area is always expressed in square units. Common units include square centimeters (cm²), square meters (m²), and square millimeters (mm²), depending on the units of the radius.
6. How is the area of a hollow hemisphere calculated?
For a hollow hemisphere, you need to find the difference between the total surface area of the outer hemisphere and the total surface area of the inner hemisphere. Let r1 be the inner radius and r2 be the outer radius. The formula becomes: TSAhollow = 3π(r2² - r1²)
7. What are some real-world applications of calculating the area of a hemisphere?
Calculating the surface area of a hemisphere is useful in various applications, including: * Determining the amount of paint needed to cover a dome-shaped structure. * Calculating the material required for manufacturing hemispherical bowls or containers. * Estimating the surface area of a planet's hemisphere for scientific studies.
8. What are common mistakes students make when calculating the surface area of a hemisphere?
Common mistakes include: * Confusing the formula for curved surface area with the formula for total surface area. * Forgetting to square the radius in the formula. * Using incorrect units for the answer (not square units).
9. If the radius of a hemisphere is doubled, how does this affect its total surface area?
If the radius (r) is doubled, the total surface area (3πr²) will increase by a factor of four (because (2r)² = 4r²).
10. Can I use the same formulas for a solid hemisphere and a hollow hemisphere?
No. For a solid hemisphere, you use the standard formulas for curved surface area (2πr²) and total surface area (3πr²). For a hollow hemisphere, you must calculate the difference between the outer and inner surface areas.
11. How does the formula for the surface area of a hemisphere relate to the formula for the surface area of a sphere?
The curved surface area of a hemisphere (2πr²) is exactly half the surface area of a full sphere (4πr²). The total surface area of a hemisphere includes the additional area of its circular base.
12. What is the relationship between the area and the volume of a hemisphere?
While not directly related through a simple formula, both area and volume calculations for a hemisphere depend on its radius (r). The volume is (2/3)πr³, which is directly influenced by the radius's cube. The surface area is affected by the square of the radius.
