The Curved Surface Area (CSA) of a hemisphere is exactly half the surface area of a full sphere. Since the total surface area of a sphere is 4πr², the curved surface area of a hemisphere is 2πr². The Total Surface Area (TSA) of a hemisphere adds the circular base, giving TSA = 3πr².
CSA of hemisphere = 2πr², where r is the radius.
TSA of hemisphere = 3πr² (includes flat circular base of area πr²).
The curved surface of a hemisphere is exactly half the surface area of a full sphere (4πr²).
Volume of hemisphere = (2/3)πr³.
If the radius is doubled, the CSA becomes 4 times larger (since area ∝ r²).
TSA/CSA ratio for a hemisphere is 3:2.
Archimedes first proved the surface area of a sphere equals 4πr².
A hemisphere is exactly half of a sphere, cut along a great circle (a plane through the center).
Surface area of a full sphere = 4πr² Curved Surface Area of hemisphere = (1/2) × 4πr² = 2πr²
Where r is the radius of the hemisphere.
The curved surface area covers only the dome-shaped (rounded) part and does NOT include the flat circular base.
The Total Surface Area (TSA) of a hemisphere includes both:
TSA = 2πr² + πr² = 3πr²
Where r is the radius of the hemisphere.
Summary:
The surface area of a sphere of radius r is 4πr². This result was proved by Archimedes, who showed that the lateral surface of a sphere equals the lateral surface of the circumscribed cylinder.
When a sphere is cut into two equal halves along a diameter:
This derivation assumes a solid or hollow hemisphere where the flat face is included as part of the surface.
Example 1: Find the CSA and TSA of a hemisphere with radius = 7 cm. CSA = 2πr² = 2 × (22/7) × 7² = 2 × (22/7) × 49 = 2 × 154 = 308 cm² TSA = 3πr² = 3 × (22/7) × 49 = 3 × 154 = 462 cm²
Example 2: Find the CSA of a hemisphere with diameter = 10 cm. r = 10/2 = 5 cm CSA = 2πr² = 2 × π × 25 = 50π ≈ 157.08 cm²
Example 3: If CSA of a hemisphere = 1386 cm², find its radius. 2πr² = 1386 2 × (22/7) × r² = 1386 (44/7) × r² = 1386 r² = 1386 × 7 / 44 = 9702 / 44 = 220.5 → r ≈ 14.85 cm
Alternative: 2 × (22/7) × r² = 1386 → r² = 1386 × 7 / 44 = 220.5, r ≈ 14.85 cm
Example 4: A bowl is in the shape of a hemisphere of radius 21 cm. Find the inner curved surface area. CSA = 2πr² = 2 × (22/7) × 21² = 2 × (22/7) × 441 = 2 × 1386 = 2772 cm²
Understanding how hemisphere areas relate to a full sphere:
| Shape | Surface Area |
|---|---|
| Full Sphere | 4πr² |
| Hemisphere CSA (dome only) | 2πr² |
| Hemisphere TSA (dome + base) | 3πr² |
Key observations:
Common applications: bowls, domes of buildings, satellite dishes, and half-sphere containers are all modelled as hemispheres.
The curved surface area (CSA) of a hemisphere is 2πr², where r is the radius. This is exactly half the surface area of a full sphere (which is 4πr²). It covers only the dome-shaped curved part, not the flat circular base.
The total surface area (TSA) of a hemisphere is 3πr². It is the sum of the curved surface area (2πr²) and the area of the flat circular base (πr²), giving TSA = 2πr² + πr² = 3πr².
CSA = 2πr² = 2 × (22/7) × 14² = 2 × (22/7) × 196 = 2 × 616 = 1232 cm². TSA = 3πr² = 3 × (22/7) × 196 = 3 × 616 = 1848 cm².
CSA (Curved Surface Area) = 2πr² covers only the rounded dome part. TSA (Total Surface Area) = 3πr² covers the dome plus the flat circular base (πr²). Their difference is πr², which is the area of the circular base.
The volume of a hemisphere = (2/3)πr³, where r is the radius. It is exactly half the volume of a full sphere (which is 4πr³/3).
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