Spheres and hemispheres
Separate a hemisphere's curved surface from its circular cut face.
Core and Extended.
Before you begin
- Square and cube a radius.
- Distinguish a hemisphere from a complete sphere.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is 4³?
What you will learn
- Calculate sphere volume and area.
- Calculate hemisphere volume, curved area and total area.
Sphere formulas use one radius
A sphere has volume 4πr³/3 and surface area 4πr². For r = 3 cm these are 36π cm³ and 36π cm². Their numerical coefficients happen to match here, but their dimensions and meanings differ. A supplied diameter must be halved before squaring or cubing.
To recover a radius from area, use √(A/(4π)); from volume, use the positive cube root of 3V/(4π). Rounding an intermediate radius can distort later area and volume results. Keep exact values or calculator precision until the final requested measure.
Cutting creates a new boundary
A hemisphere has half the sphere's volume, 2πr³/3, and half its curved surface area, 2πr². Its flat circular base has area πr². A closed hemisphere's total area is therefore 3πr², not half the complete sphere area alone.
A hemispherical bowl open at the top needs curved area only when wall thickness is ignored. A solid hemisphere includes the flat cut face unless it is joined to another component. For r = 4 cm, curved area is 32π cm², total area 48π cm² and volume 128π/3 cm³. Identify which face is exposed in the context.
Pause and explain
A closed hemisphere has radius 3 cm. Find total area.
Worked example
A closed solid hemisphere has radius 4 cm. Find volume and total area.
Show the worked solution
- Volume is half a sphere: (2/3)π × 4³ = 128π/3 cm³.
- The curved surface area is 2π × 4² = 32π cm².
- Add the flat base 16π cm² to obtain total area 48π cm².
Answer Volume 128π/3 cm³; total area 48π cm².
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Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find exact area and volume of a sphere with radius 6 cm.
Give me a hint
Square for area and cube for volume.
Compare my reasoning
- Area is 4π × 36 = 144π cm².
- Volume is (4/3)π × 216 = 288π cm³.
- Keep the unit powers distinct.
144π cm²; 288π cm³.
Look for these in your work
- Used the right radius powers.
- Kept exact pi answers.
A sphere has area 100π cm². Find its radius and volume.
Give me a hint
First solve 4πr² = 100π.
Compare my reasoning
- r² = 25, so radius is 5 cm.
- Volume is (4/3)π × 125 = 500π/3 cm³.
- Check the recovered radius in the area formula.
Radius 5 cm; volume 500π/3 cm³.
Look for these in your work
- Recovered a positive radius.
- Used it consistently in volume.
An open hemispherical bowl has radius 5 cm. Find its inner curved area and ideal capacity, ignoring thickness.
Give me a hint
The open circular rim is not a filled base face.
Compare my reasoning
- Curved area is 2π × 25 = 50π cm².
- Capacity is (2/3)π × 125 = 250π/3 cm³.
- Do not add πr² to the area because the opening is not covered.
Curved area 50π cm²; capacity 250π/3 cm³.
Look for these in your work
- Excluded the open face.
- Halved the sphere volume.
Common mistakes
- Halving sphere area and forgetting the flat face.
- Using diameter as radius.
- Confusing surface area with capacity.
What can you explain now?
What is added to hemisphere curved area for a closed solid?
Compare with the explanation
One circular base of area πr².
The cut introduces a flat face, so total area is 2πr² + πr² = 3πr².
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Make it stick
Tomorrow, compare a closed hemisphere and an open hemispherical bowl of the same radius, explaining their different areas.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.