Surface Area of Hemisphere
A hemisphere is exactly half of a sphere, obtained by cutting a sphere along a great circle (a circle passing through the centre). It has one curved surface and one flat circular face.
Unlike a full sphere, a hemisphere requires two separate surface area formulas: the Curved Surface Area (CSA) for the dome, and the Total Surface Area (TSA) that includes the flat base.
Hemispheres appear in everyday objects — bowls, domes, igloos, and half-watermelons. This topic is part of NCERT Class 9 Mathematics, Chapter: Surface Areas and Volumes.
What is Surface Area of Hemisphere?
Definition: A hemisphere is half of a sphere, bounded by a great circle and the curved surface above (or below) it.
Curved Surface Area (CSA):
CSA = 2πr²
Total Surface Area (TSA):
TSA = 3πr²
Where:
- r = radius of the hemisphere (same as the original sphere)
- CSA = area of the curved dome surface only
- TSA = CSA + area of the flat circular base (πr²)
Important:
- CSA of hemisphere = half the surface area of the full sphere = (1/2) × 4πr² = 2πr²
- TSA = CSA + base area = 2πr² + πr² = 3πr²
- Use CSA when only the curved surface is painted or coated.
- Use TSA when the entire outer surface (including the base) needs covering.
Surface Area of Hemisphere Formula
Key Formulas:
1. Curved Surface Area of hemisphere:
CSA = 2πr²
2. Total Surface Area of hemisphere:
TSA = 3πr²
3. Base area:
- Area of flat circular base = πr²
4. In terms of diameter (d = 2r):
- CSA = 2π(d/2)² = πd²/2
- TSA = 3π(d/2)² = 3πd²/4
5. Finding radius from CSA:
- r² = CSA / (2π)
- r = √[CSA / (2π)]
6. Finding radius from TSA:
- r² = TSA / (3π)
- r = √[TSA / (3π)]
7. Relationship:
- TSA = (3/2) × CSA
- CSA = (2/3) × TSA
Derivation and Proof
Derivation of Surface Area of Hemisphere:
Step 1: Start with the full sphere
- Surface area of a full sphere = 4πr²
Step 2: Cut the sphere in half
- When a sphere is cut along a great circle, each half is a hemisphere.
- The curved surface of each hemisphere = (1/2) × 4πr² = 2πr²
Step 3: Identify the flat face
- The cut creates a flat circular face with radius r.
- Area of the flat face = πr²
Step 4: Calculate CSA
- CSA = curved part only = 2πr²
Step 5: Calculate TSA
- TSA = CSA + Base area
- TSA = 2πr² + πr²
- TSA = 3πr²
Verification:
- CSA of hemisphere (2πr²) = 2 × area of one great circle (πr²)
- TSA of hemisphere (3πr²) = 3 × area of one great circle (πr²)
- TSA of hemisphere = (3/4) × surface area of full sphere
Types and Properties
Types of hemisphere surface area problems:
1. CSA only
- Use CSA = 2πr²
- Applies when only the dome/curved part is to be painted, coated, or covered.
- Example: painting the inner curved surface of a bowl.
2. TSA (solid hemisphere)
- Use TSA = 3πr²
- Applies when the entire surface including the flat base is exposed.
- Example: polishing a decorative solid hemisphere.
3. Hollow hemisphere
- Outer CSA = 2πR²
- Inner CSA = 2πr²
- Rim (annular ring) area = π(R² − r²)
- Total exposed area = 2πR² + 2πr² + π(R² − r²)
4. Cost-based problems
- Find the required surface area (CSA or TSA).
- Multiply by the cost per unit area.
5. Comparison problems
- Comparing CSA and TSA of hemispheres of different radii.
- Comparing hemisphere surface area with that of a sphere, cylinder, or cone.
Solved Examples
Example 1: Example 1: CSA of a 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: CSA = 1232 cm².
Example 2: Example 2: TSA of a solid hemisphere
Problem: Find the total surface area of a solid hemisphere of radius 7 cm. (Use π = 22/7)
Solution:
Given:
- r = 7 cm
Using TSA = 3πr²:
- TSA = 3 × (22/7) × 49
- TSA = 3 × 22 × 7
- TSA = 462 cm²
Answer: TSA = 462 cm².
Example 3: Example 3: Finding radius from CSA
Problem: The curved surface area of a hemisphere is 693 cm². Find the radius. (Use π = 22/7)
Solution:
Given:
- CSA = 693 cm²
Using CSA = 2πr²:
- 693 = 2 × (22/7) × r²
- 693 = (44/7) × r²
- r² = 693 × 7 / 44 = 4851/44 = 110.25
- r = √110.25 = 10.5 cm
Answer: Radius = 10.5 cm.
Example 4: Example 4: Cost of painting a dome
Problem: A hemispherical dome has a radius of 21 m. Find the cost of painting its inner curved surface at Rs 10 per m². (Use π = 22/7)
Solution:
Given:
- r = 21 m, rate = Rs 10 per m²
CSA of dome:
- CSA = 2 × (22/7) × 21²
- CSA = 2 × (22/7) × 441
- CSA = 2 × 22 × 63
- CSA = 2772 m²
Cost:
- Cost = 2772 × 10 = Rs 27,720
Answer: Cost of painting = Rs 27,720.
Example 5: Example 5: TSA given diameter
Problem: Find the total surface area of a solid hemisphere of diameter 28 cm. (Use π = 22/7)
Solution:
Given:
- d = 28 cm, so r = 14 cm
Using TSA = 3πr²:
- TSA = 3 × (22/7) × 196
- TSA = 3 × 616
- TSA = 1848 cm²
Answer: TSA = 1848 cm².
Example 6: Example 6: Comparing CSA and TSA
Problem: Find the ratio of CSA to TSA of a hemisphere.
Solution:
CSA = 2πr²
TSA = 3πr²
Ratio:
- CSA : TSA = 2πr² : 3πr² = 2 : 3
Answer: The ratio of CSA to TSA is 2 : 3.
Example 7: Example 7: Hemisphere vs sphere surface area
Problem: Show that the TSA of a hemisphere is (3/4) of the surface area of the full sphere.
Solution:
SA of sphere = 4πr²
TSA of hemisphere = 3πr²
Ratio:
- TSA / SA = 3πr² / 4πr² = 3/4
- TSA = (3/4) × SA of sphere
Answer: Verified. TSA of hemisphere = (3/4) × surface area of sphere.
Example 8: Example 8: Finding radius from TSA
Problem: The total surface area of a solid hemisphere is 462 cm². Find its radius. (Use π = 22/7)
Solution:
Given:
- TSA = 462 cm²
Using TSA = 3πr²:
- 462 = 3 × (22/7) × r²
- 462 = (66/7) × r²
- r² = 462 × 7 / 66 = 3234/66 = 49
- r = √49 = 7 cm
Answer: Radius = 7 cm.
Example 9: Example 9: Hollow hemisphere
Problem: A hollow hemisphere has an outer radius of 10 cm and inner radius of 8 cm. Find the total surface area exposed. (Use π = 3.14)
Solution:
Given:
- R = 10 cm (outer), r = 8 cm (inner)
Total exposed area:
- Outer CSA = 2πR² = 2 × 3.14 × 100 = 628 cm²
- Inner CSA = 2πr² = 2 × 3.14 × 64 = 401.92 cm²
- Rim (annular ring) = π(R² − r²) = 3.14 × (100 − 64) = 3.14 × 36 = 113.04 cm²
- Total = 628 + 401.92 + 113.04 = 1142.96 cm²
Answer: Total exposed surface area = 1142.96 cm².
Example 10: Example 10: Hemisphere placed on cylinder
Problem: A solid toy is in the shape of a hemisphere surmounted on a cylinder. The radius of the hemisphere and the cylinder is 7 cm, and the height of the cylinder is 10 cm. Find the TSA of the toy. (Use π = 22/7)
Solution:
Given:
- r = 7 cm (common radius), h = 10 cm (cylinder height)
TSA of the toy:
- CSA of hemisphere = 2πr² = 2 × (22/7) × 49 = 308 cm²
- CSA of cylinder = 2πrh = 2 × (22/7) × 7 × 10 = 440 cm²
- Base of cylinder = πr² = (22/7) × 49 = 154 cm²
- TSA = 308 + 440 + 154 = 902 cm²
Answer: TSA of the toy = 902 cm².
Real-World Applications
Applications of Surface Area of Hemisphere:
- Architecture: Calculating the area of hemispherical domes for painting, roofing, or waterproofing (e.g., planetariums, mosques, stadiums).
- Cooking: Estimating the capacity and material for hemispherical bowls and woks.
- Construction: Material estimation for hemispherical water tanks and storage vessels.
- Design: Manufacturing hemispherical lampshades, helmets, and decorative items.
- Geography: The Earth is divided into Northern and Southern hemispheres; surface area calculations apply to mapping and climate studies.
- Science: Calculating surface areas of hemispherical lenses and reflectors in optics.
Key Points to Remember
- CSA of hemisphere = 2πr² (curved surface only).
- TSA of hemisphere = 3πr² (curved surface + flat circular base).
- Base area of hemisphere = πr².
- CSA of hemisphere = half the surface area of the full sphere.
- TSA of hemisphere = (3/4) of the surface area of the full sphere.
- Ratio of CSA : TSA = 2 : 3.
- TSA = (3/2) × CSA.
- For hollow hemispheres, include outer CSA, inner CSA, and the annular ring at the rim.
- Surface area is in square units (cm², m²).
- Use CSA for painting/coating the dome; use TSA when the base is also exposed.
Practice Problems
- Find the CSA of a hemisphere of radius 21 cm. (Use π = 22/7)
- Find the TSA of a solid hemisphere of radius 3.5 cm.
- The CSA of a hemisphere is 308 cm². Find its radius. (Use π = 22/7)
- A hemispherical bowl of radius 10.5 cm is to be painted on the inside. Find the area to be painted.
- The TSA of a hemisphere is 1848 cm². Find its diameter.
- A solid hemisphere is placed on a table with its flat face down. Its radius is 5 cm. Find the area of the surface exposed to air.
- A hollow hemispherical shell has outer radius 7 cm and inner radius 5 cm. Find the total area that needs polishing.
- The ratio of CSA to base area of a hemisphere is 2 : 1. Verify this.
Frequently Asked Questions
Q1. What is the difference between CSA and TSA of a hemisphere?
CSA = 2πr² counts only the curved dome surface. TSA = 3πr² includes both the curved surface and the flat circular base.
Q2. Why is the CSA of a hemisphere 2πr²?
The total surface area of a full sphere is 4πr². A hemisphere is half a sphere, so its curved surface area = (1/2) × 4πr² = 2πr².
Q3. When do we use CSA vs TSA?
Use CSA when only the dome is painted or coated (like the inside of a bowl). Use TSA when the entire surface including the flat base needs covering (like polishing a solid hemisphere paperweight).
Q4. What is the ratio of CSA to TSA of a hemisphere?
CSA : TSA = 2πr² : 3πr² = 2 : 3.
Q5. How do you find the radius from the TSA of a hemisphere?
Use r² = TSA / (3π), then take the square root. For example, if TSA = 462 cm²: r² = 462 × 7 / 66 = 49, so r = 7 cm.
Q6. What is the surface area of a hollow hemisphere?
Total exposed area = outer CSA (2πR²) + inner CSA (2πr²) + annular ring (π(R² − r²)), where R and r are outer and inner radii.
Q7. Is hemisphere surface area in the CBSE Class 9 syllabus?
Yes. Both CSA and TSA of hemisphere are covered in CBSE Class 9 Mathematics, Chapter: Surface Areas and Volumes.
Q8. How does the TSA of a hemisphere compare to the sphere's surface area?
TSA of hemisphere (3πr²) is exactly (3/4) of the surface area of the full sphere (4πr²). The additional πr² comes from the flat base.
Related Topics
- Surface Area of Sphere
- Volume of Hemisphere
- Surface Area of Cone
- Surface Area of Cylinder
- Volume of Cone
- Volume of Sphere
- Slant Height of Cone
- Combination of Solids
- Conversion of Solids
- Frustum of a Cone
- Surface Area of Combined Solids
- Volume Word Problems
- Volume of Frustum of Cone
- Curved Surface Area of Frustum
