If the total surface area of a solid hemisphere is equal to its volume, find the radius

Analyze the question: A solid hemisphere has a total surface area (curved surface plus the flat circular base). We need the radius r at which this area equals the volume.

Concept and formulas:

• Curved surface area = 2πr²
• Flat circular base = πr²
• Total surface area = 3πr²
• Volume = (2/3)πr³

Solution:

Set the total surface area equal to the volume:

3πr² = (2/3)πr³

Divide both sides by πr²:

3 = (2/3)r

Multiply both sides by 3/2:

r = 9/2 = 4.5 cm

Check:

• Total surface area = 3π(4.5)² = 3π(20.25) = 60.75π
• Volume = (2/3)π(4.5)³ = (2/3)π(91.125) = 60.75π ✓

Answer: B (4.5 cm)

Tip: For a solid hemisphere, remember "total surface area = 3πr²", because you must add the flat base to the curved part. Many students forget the base and use 2πr², which would give r = 3 (option A), a common trap. Always check whether the question says "curved" or "total" surface area.