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.
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.
After trying it yourself, choose your next review. This is your self-assessment.
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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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources