Circle Calculator

Circle Calculator Quick Framework

Fast reference to move from any one known circle value (r, d, C, A, θ) to every other property plus arc / sector results without wading through a long article.

1. Core Relationships (memorize 90%)

• d = 2r
• C = 2πr = πd
• A = πr²
• Arc length L = (θ/360°)·C = (θ°/360°)·2πr = θ (rad) · r
• Sector area As = (θ/360°)·A = (θ/360°)·πr² = ½ θ (rad) r²

2. Solve Path (start from what you know)

1. Known radius r → compute d, C, A directly.
2. Known diameter d → r = d/2 → then C, A.
3. Known circumference C → r = C/(2π) → then d, A.
4. Known area A → r = √(A/π) → then d, C.
5. Arc / sector given θ + r (or any converted r) → apply L / As formulas.

3. Quality / Sanity Checks

• Units: linear (r,d,C,L) vs squared (A, As).
• θ bounds: 0 < θ ≤ 360° (or 0 < θ ≤ 2π rad).
• Arc proportion: L/C = θ/360°. Sector proportion: As/A = θ/360°.
• If θ = 180° → L = C/2, As = A/2 (quick half check).

4. Shortcuts & Mental Hacks

• C ≈ 6.283r (2π ≈ 6.283). For rough: C ≈ 3.14d.
• A scales with r²: doubling r → area ×4, circumference ×2.
• Quarter circle (90°): L = C/4, As = A/4.
• θ in radians quickly: θ° × π/180.

5. Pitfalls

• Using diameter where radius required (area, arc, sector).
• Mixing degrees with radian formulas (½ θ r² needs θ in radians).
• Premature rounding of r before computing A or sector.
• Negative or zero inputs (reject).

6. Minimal Conversion Table

Have C? r = C/(2π) → A = C²/(4π)

Have A? r = √(A/π) → C = 2√(πA)

Have L & θ°? r = (180·L)/(π·θ°)

Have sector area As & θ°? r = √( (360·As)/(θ°·π) )

7. Quick Validation Example

Input r = 5 cm, θ = 90°. Expect C ≈ 31.416, A ≈ 78.540, L ≈ 7.854, As ≈ 19.635. Ratios L/C ≈ 0.25, As/A ≈ 0.25 → passes.

8. FAQ (Micro)

Best starting value? Radius or diameter—fewest steps.

When to use radians? Prefer for calculus or when θ given in rad directly (arc = θr, sector = ½θr² is cleaner).

Precision tip? Keep at least 4–5 decimals for r before deriving A.

9. Action Tip

Always normalize to radius first; every other circle property flows from r with a single substitution.