lesson
Distinguish surface and volume
1 · Learn
A sphere of radius r has surface area 4πr² and volume 4πr³/3. Squared and cubed units distinguish the quantities. Diameter must first be halved if the formula expects radius.
For a hemisphere, a closed surface includes a flat circular base as well as half the sphere's curved surface. Read whether the base is included.
2 · Worked example
A sphere of radius 3 cm has surface area 36π cm² and volume 36π cm³; the equal coefficients do not make area and volume the same quantity.
3 · Your turn
Find the volume of a sphere of radius 2 cm in exact form.
Check your answer
32π/3 cm³.
4 · Apply your learning
Explain how surface area and volume scale when the radius doubles.
Adult guidance and safety
Check prerequisites, units and reasoning. Provide accurate diagrams and varied practice. Extension content needs suitable prior understanding; no GCSE board, tier or examination readiness is implied by this proposed sequence.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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