A hemisphere is simply half of a sphere, but its surface area includes both a curved outer part and a flat circular base. Understanding its surface area helps in solving geometry problems and real-world applications like designing domes, bowls, satellite dishes, helmets, and containers. Whether you’re preparing for CUET, SSC, JEE, or school exams, knowing the TSA and CSA formulas of a hemisphere is essential for fast and accurate calculations.
Surface Area of a Hemisphere – Formula Overview
| Formula | Variables & Meaning | When It Is Used |
|---|---|---|
| TSA = 3πr² | r = radius, π = 3.1416 | Used to find total surface area (curved part + base) |
| CSA = 2πr² | r = radius, π = 3.1416 | Used when only the curved surface area is needed |
What is the Surface Area of a Hemisphere?
In mathematics, the surface area of a hemisphere measures the total area covering half of a sphere. It includes the curved region and sometimes the flat circular base. A hemisphere has two components:
- Curved Surface Area (CSA) – This is half of the sphere’s outer surface, given by 2πr².
- Base Area – A flat circle at the bottom, given by πr².
Adding these together gives the Total Surface Area (TSA):
TSA = CSA + Base Area = 2πr² + πr² = 3πr²
These formulas help calculate material costs, construction measurements, and surface exposure in various real-life applications such as domes, dishes, helmets, and tanks.
Examples to Calculate Surface Area of a Hemisphere
Example 1: Find the TSA of a hemisphere with radius 7 cm
Step 1: TSA = 3πr²
Step 2: = 3 × 3.1416 × 7²
Step 3: = 3 × 3.1416 × 49
Result: 461.81 cm²
So, the total surface area of the hemisphere is 461.81 cm².
Example 2: Find the CSA of a hemisphere with radius 10 m
Step 1: CSA = 2πr²
Step 2: = 2 × 3.1416 × 10²
Step 3: = 2 × 3.1416 × 100
Result: 628.32 m²
So, the curved surface area of the hemisphere is 628.32 m².
FAQs about Surface Area of a Hemisphere Formula
Q1. What is the difference between TSA and CSA of a hemisphere?
TSA includes both the curved surface and the flat base, while CSA includes only the curved part.
Q2. Why does TSA include a multiplication by 3?
Because TSA = 2πr² (curved) + πr² (base) = 3πr².
Q3. What units are used for surface area?
Surface area is always measured in square units like cm², m², or mm².
Q4. Can we find surface area without including the base?
Yes, that is the curved surface area (CSA), given by 2πr².
Q5. Where is this formula used in practical applications?
In designing domes, helmets, containers, bowls, and any half-spherical structures.
Q6. Is this formula important for competitive exams?
Yes, it is commonly tested in CUET, JEE, SSC, board exams, and other geometry-based tests.