Core answer: Two formulas run every circle problem: circumference C = 2πr = πd and area A = πr². A 10-inch pizza (r = 5 in) has area 25π ≈ 78.5 in²; a 12-inch is 36π ≈ 113 in² — 44% more pizza for usually 20% more money. Use π ≈ 3.14159; for quick mental math 3.14 or 22/7 is fine.
The core formulas
| Quantity | Formula | Example (r = 5) |
|---|---|---|
| Circumference | C = 2πr | 31.42 |
| Area | A = πr² | 78.54 |
| Arc length (θ in degrees) | L = 2πr × θ/360 | 90° arc → 7.85 |
| Sector area | A = πr² × θ/360 | 90° sector → 19.63 |
| Segment (chord + arc) | sector − triangle | engineering |
Why area grows with the SQUARE of radius
Double the radius and the area quadruples; triple it and you get 9×. This single fact explains pizza pricing (a 16-inch is 4× a 8-inch), shower-head flow, and why doubling pipe diameter moves 4× the water.
Pizza example. Two 10-inch pizzas (157 in² total) cost more than one 14-inch (154 in²) at many shops — nearly the same area, worse price. Buy the big one.
Worked examples
Example 1 — Fencing a round garden. Radius 3 m: C = 2π×3 ≈ 18.85 m of edging. Grass seed: A = π×9 ≈ 28.3 m²; at 25 g/m² you need ~0.7 kg.
Example 2 — Wheel distance. A 26-inch bike wheel (r = 33 cm): each turn covers 2π×0.33 ≈ 2.07 m; 1 km is ~483 revolutions.
Example 3 — Cable reel. A reel of inner radius 0.3 m, outer 0.6 m: annulus area = π(0.6² − 0.3²) = 0.27π ≈ 0.85 m² — multiply by cable layers for length.
Common mistakes and myths
- Squaring the diameter — A = πr² uses the radius; A = πd²/4 if you insist on diameter. πd² is 4× too big.
- Confusing circumference and area — C grows linearly with r, A grows quadratically; they never share units.
- Using 3.14 everywhere in engineering — fine to 0.05%, but high-precision CNC uses full π from the calculator.
- Forgetting arc vs chord — the arc is the curved path (longer); the chord is the straight shortcut.
- Assuming π is exactly 22/7 — 22/7 = 3.14286, off by 0.04%; never use it in code.