Circle measurement and 3D solids
The "circle and curved-solid" formulae are essential for GCSE — many appear on the formula sheet, but it pays to know them by heart.
Circle
- Circumference = 2πr = πd.
- Area = πr².
Worked example: circle radius 5 cm.
- Circumference = 10π ≈ 31.42 cm.
- Area = 25π ≈ 78.54 cm².
Sphere
- Surface area = 4πr².
- Volume = (4/3) πr³.
Worked example: sphere radius 6 cm.
- SA = 4 × π × 36 = 144π ≈ 452.4 cm².
- V = (4/3) × π × 216 = 288π ≈ 904.8 cm³.
Cylinder
- Surface area = 2πr² + 2πrh = 2πr(r + h).
- Volume = πr²h.
Cone
- Curved surface area = πrℓ (where ℓ = slant height).
- Total surface area = πr² + πrℓ.
- Volume = (1/3) πr²h.
Worked example: cone radius 4 cm, slant height 5 cm, perpendicular height 3 cm.
- Curved SA = 20π ≈ 62.83 cm².
- Volume = (1/3) × π × 16 × 3 = 16π ≈ 50.27 cm³.
Pyramid
- Volume = (1/3) × base area × height.
Worked example: square-based pyramid with base 6 × 6 cm and height 9 cm.
- V = (1/3) × 36 × 9 = 108 cm³.
Slant vs perpendicular height for cones
In a cone, the slant height ℓ relates to the perpendicular height h and radius r by Pythagoras: ℓ² = r² + h².
⚠Common mistakes
- Using radius squared in circumference — circumference uses r (linear), area uses r².
- Slant height vs perpendicular height — pick the right one!
- Surface area vs volume formulas — keep units in mind: cm² for SA, cm³ for V.
- Forgetting (1/3) factor in cone and pyramid volume.
- Forgetting both end faces in cylinder surface area.
➜Try this— Quick check
A cone has radius 3 cm, slant height 5 cm. Find the curved surface area in terms of π.
- Curved SA = πrℓ = π × 3 × 5 = 15π cm².
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