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Surface Area of a Hemisphere Formula

The hemisphere surface area formulas express curved shell area and full exterior area using the radius.

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Hemisphere diagram and geometry notes for surface area guides

Quick Answer

CSA = 2πr². TSA = 3πr². Base = πr².

Formula

  • CSA = 2πr²
  • TSA = 3πr²
  • Base = πr²

Introduction

Surface Area of a Hemisphere Calculator applies CSA = 2πr² and TSA = 3πr² whenever you enter radius, diameter, or a related value.

Students often memorize the symbols before they understand why the curved part is 2πr² instead of πr². This article connects each formula to the shape.

You will also see how radius scaling affects both curved and total results on engineering sketches.

Main Content

Geometric interpretation and radius relationship

Think of CSA as half of the orange peel on a full sphere. Think of TSA as that peel plus the flat slice face.

Radius is the single dimension that drives every result. Diameter d relates through r = d/2.

Some sheets list total surface area directly. You can solve r = √(TSA / (3π)) when you need to back into a radius from a coating spec.

For dome-only coatings, compare your result with the curved surface area guide so you do not add the base by habit.

  • CSA: use when the flat base is not part of the exterior count
  • TSA: use when the full outside of a solid hemisphere is required
  • Always square the same length unit for area

Curved and total surface area formulas

  • CSA = 2πr²
  • TSA = 2πr² + πr² = 3πr²

A sphere of radius r has surface area 4πr². Cutting it in half leaves a curved cap with half that shell: 2πr².

The flat cut is a circle with area πr². Total surface area adds the cap and base: 2πr² + πr² = 3πr².

If you double the radius, each area quadruples because r is squared. That scaling rule matters when comparing two dome sizes on a bid.

New to the vocabulary? Start with what hemisphere surface area means before you rearrange formulas for homework.

Worked formula example

Given r = 6 in, CSA = 2π(36) = 72π ≈ 226.19 in².

TSA = 3π(36) = 108π ≈ 339.29 in².

The base alone is 36π ≈ 113.10 in², which is exactly the difference between TSA and CSA.

FAQ

Why is curved area 2πr² and not πr²?
πr² is the flat base. The curved cap is half of 4πr², which equals 2πr².
Can I write the formulas using diameter?
Yes. With d = 2r, CSA = (πd²)/2 and TSA = (3πd²)/4.

Conclusion

Use 2πr² for the curved cap and 3πr² when the base belongs in the total exterior.

Match the formula to the wording of the task, then verify on the home calculator.

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