Triangle Calculator Quick Framework
Minimal decision map: start from your known sides / angles, validate the triangle, derive everything else (area, remaining sides, angles, radii, heights) fast.
1. Validity First
- Triangle inequality: a + b > c; a + c > b; b + c > a.
- Angles must sum to 180° (within rounding).
- All sides > 0, all angles between (0°, 180°).
2. Primary Solve Routes
- SSS: All sides → angles via Law of Cosines.
- SAS: Two sides + included angle → third side via Law of Cosines.
- ASA / AAS: Two angles + one side → remaining angle (180° sum) then Law of Sines.
- Base + height → area direct (A = ½ b h).
- Coordinates (x1,y1),(x2,y2),(x3,y3) → side lengths by distance formula → proceed as SSS.
3. Core Formulas
- Perimeter P = a + b + c; semi-perimeter s = P/2.
- Heron: A = √[s(s-a)(s-b)(s-c)].
- Law of Cosines: c² = a² + b² - 2ab cos C.
- Law of Sines: a/sin A = b/sin B = c/sin C.
- Right triangle only: a² + b² = c².
- Inradius r = A / s.
- Circumradius R = (a b c)/(4A).
- Height to side a: ha = 2A / a (similar for b, c).
4. Fast Classification
- By sides: Equilateral (a≈b≈c), Isosceles (two equal), Scalene (all diff).
- By angles: Right (one 90°), Obtuse (>90°), Acute (all <90°).
5. Sanity Checks
- Angle sum ~ 180° (allow ≤0.1° rounding error).
- Largest angle opposite longest side.
- For right classification: |a² + b² - c²| small tolerance.
- Area positive > 0; r ≤ R always.
6. Shortcuts & Patterns
- 30-60-90 ratio 1:√3:2 (hypotenuse = 2·short leg).
- 45-45-90 ratio 1:1:√2.
- Equilateral: A = (√3/4)a²; r = a√3/6; R = a√3/3.
7. Error Pitfalls
- Using Law of Sines in ambiguous SSA directly (not supported; need logic for two solutions).
- Rounding intermediate angles early → drift in final angle sum.
- Supplying non-included angle for SAS then misusing cosine law.
8. Minimal Workflow Example (SSS 3,4,5)
Compute s = 6 → A = √[6·3·2·1] = 6. Angles: cos A = (4²+5²-3²)/(2·4·5)=0.8 → A≈36.87°. Similarly B≈53.13°, C=90°. r = A/s = 1. R = (3·4·5)/(4·6)=2.5. Checks pass.
9. FAQ (Micro)
Best universal area method? Heron when all sides known; otherwise choose method matching given data.
Detect impossible inputs? Fail triangle inequality or angle sum ≠180° → reject.
Prefer radians? Use for advanced trig; core relations unchanged.
10. Action Tip
Normalize the triangle: compute all sides first if possible (SSS reduction) then derive angles, area, radii—one clean path, fewer mistakes.