Surface Area of Hemisphere Complete Concept and Formula

The concept of surface area of hemisphere plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding how to find the curved surface area and the total surface area of a hemisphere helps students solve a wide variety of practical mensuration problems asked in school tests and competitive exams.

What Is Surface Area of Hemisphere?

A surface area of hemisphere refers to the measurement of the area covering both the curved part and the flat base (if included) of a half-sphere. A hemisphere is simply half of a sphere, formed by slicing a sphere along its center. You’ll find this concept applied in areas such as solid geometry, construction and architecture (like domes), and Physics when calculating the exposed area of a hemispherical object (like bowls, globes, and certain antennas).

Key Formula for Surface Area of Hemisphere

Here’s the standard formula for the surface area of a solid hemisphere with radius \( r \):
Curved surface area (CSA): \( 2\pi r^2 \)
Total surface area (TSA): \( 3\pi r^2 \) (includes base circle)

For a hollow hemisphere with outer radius \( R \) and inner radius \( r \), the formula is:
TSA: \( 3\pi R^2 + \pi r^2 \)

Cross-Disciplinary Usage

Surface area of hemisphere is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. For example, in Physics you may need to find the heat loss or paint required for a hemispherical shell. Students preparing for JEE or NEET will see its relevance in various mensuration and application-based questions.

Step-by-Step Illustration

1. Suppose the radius of a hemisphere is 7 cm. Find its curved and total surface area. (Take π = 22/7)
1. Given, \( r = 7 \) cm

2. Curved surface area = \( 2\pi r^2 = 2 \times \frac{22}{7} \times 7^2 = 2 \times \frac{22}{7} \times 49 = 2 \times 154 = 308 \) cm²

3. Total surface area = \( 3\pi r^2 = 3 \times \frac{22}{7} \times 49 = 3 \times 154 = 462 \) cm²

Difference Between CSA and TSA (Quick Table)

How to Derive the Formula?

1. Start from the surface area of sphere = \( 4\pi r^2 \).


2. The hemisphere is half of a sphere: so, curved surface area = \( \frac{1}{2} \times 4\pi r^2 = 2\pi r^2 \).


3. Add the base (circle): area = \( \pi r^2 \).


4. Total Surface Area = Curved + Base = \( 2\pi r^2 + \pi r^2 = 3\pi r^2 \).

Solved Examples

1. A hemispherical bowl has a radius of 10 cm. Find the total surface area.
1. Given, \( r = 10 \) cm

2. TSA = \( 3\pi r^2 = 3 \times 3.14 \times 100 = 942 \) cm²


2. The outer radius of a hollow hemispherical shell is 8 cm, inner radius is 6 cm. Find its total surface area.
1. TSA = \( 3\pi R^2 + \pi r^2 = 3 \times 3.14 \times 64 + 3.14 \times 36 = 602.88 + 113.04 = 715.92 \) cm²


3. The total surface area of a hemisphere is 616 cm². Find its radius. (Use π = 22/7)
1. TSA = 3πr² = 616

2. 3 × 22/7 × r² = 616

3. r² = (616 × 7)/(3 × 22) = 616 × 7 / 66 = 65.33

4. r ≈ 8.08 cm

Speed Trick or Vedic Shortcut

Here’s a quick shortcut that helps solve problems faster when working with surface area of hemisphere. Remember, if you know the diameter \( d \), you can quickly find CSA:

1. CSA = \( 2\pi r^2 = \frac{\pi d^2}{2} \) (since \( r = d/2 \)).
2. For TSA, add another base area: \( \frac{\pi d^2}{2} + \frac{\pi d^2}{4} = \frac{3\pi d^2}{4} \).

Tricks like these are practical in competitive exams like NTSE, Olympiads, and school boards. Vedantu’s live sessions include more such shortcuts, helping you build speed and confidence!

Try These Yourself

• Find the curved surface area of a hemisphere of radius 12 cm.
• If a hemispherical tank’s TSA is 1386 cm², what is its radius?
• A hemispherical dome has a diameter of 14 m. Calculate the paint required to cover only the curved outer surface (use 1 m² = 150 Rs).
• Compare the TSA of a hemisphere and a sphere with the same radius.

Frequent Errors and Misunderstandings

• Mistaking curved surface area (CSA) for total surface area (TSA).
  TSA always includes the base circle, CSA does not.
• Using diameter instead of radius, or vice versa. Always double-check which is given.
• Not expressing the answer in correct square units (cm², m²).
• Applying the sphere’s formula in place of a hemisphere.

Relation to Other Concepts

The idea of surface area of hemisphere connects closely with topics such as surface area of sphere and surface area of cylinder. It also links to volume of hemisphere for solving compound solid problems. Mastering this helps with advanced geometry and physics problems.

Classroom Tip

A quick way to remember CSA vs. TSA for a hemisphere: “Curved is 2, Total is 3” (think: number of parts included in the formula with πr²!). Vedantu’s teachers often use this rhyme during live classes to help students quickly recall these formulas during practice and exams.

We explored surface area of hemisphere—from definition, formula, examples, and tricks to common mistakes and topic connections. Continue practicing with Vedantu to become confident in using this concept for all types of maths questions!

Related Topics for Further Learning: Surface Area of Sphere | Volume of Hemisphere | Mensuration Formulas Class 10