Circles, spheres, cones and pyramids

Circle calculations appear on every OCR J560 paper. Higher-tier adds surface area and volume of 3D shapes. All formulae (except C = 2πr and A = πr²) are given on the OCR formula sheet — but you must know how to apply them correctly.

Circles

Circumference = 2πr = πd (where r = radius, d = diameter).

Area = πr² (r is ALWAYS the radius, not the diameter).

Example: Circle with diameter 12 cm:

• r = 6 cm.
• Circumference = 2π × 6 = 12π cm (exact) ≈ 37.7 cm (3 s.f.).
• Area = π × 6² = 36π cm² (exact) ≈ 113 cm² (3 s.f.).

Arcs and sectors (G18 links)

Arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr².

where θ is the angle at the centre in degrees.

Sphere

Volume = (4/3)πr³. Surface area = 4πr².

Example: Sphere radius 3 cm:

• Volume = (4/3)π(3³) = (4/3)π × 27 = 36π cm³ ≈ 113 cm³.
• Surface area = 4π(3²) = 36π cm² ≈ 113 cm² (coincidence!).

Cylinder (recap)

Volume = πr²h. Curved surface area = 2πrh. Total surface area = 2πrh + 2πr².

Cone

Volume = (1/3)πr²h. Curved surface area = πrl (where l = slant height = √(r² + h²)). Total surface area = πrl + πr².

Example: Cone with radius 5 cm and height 12 cm:

• Slant height l = √(5² + 12²) = √169 = 13 cm.
• Volume = (1/3)π(25)(12) = 100π cm³ ≈ 314 cm³.
• Curved SA = π × 5 × 13 = 65π cm² ≈ 204 cm².

Pyramid

Volume = (1/3) × base area × height.

Example: Square base pyramid, base 6 cm, height 4 cm:

• Volume = (1/3) × 36 × 4 = 48 cm³.

Composite shapes

Many OCR questions combine shapes (e.g., cone on a cylinder, hemisphere on a cone). Work out each part separately, then add or subtract volumes/areas.

Common OCR exam mistakes

1. Using diameter instead of radius in πr² → gives area four times too large.
2. Forgetting to halve the diameter before squaring: r = d/2.
3. Cone curved SA: using h (height) instead of l (slant height) in πrl.
4. Not including both flat circles when asked for TOTAL surface area of a cylinder.
5. Rounding π too early — use the π button on the calculator; only round the final answer.