Triangle Calculator

**Triangle from three sides (SSS):** **Heron's formula (area):** Let s = (a + b + c) / 2 (semi-perimeter). A = √(s × (s-a) × (s-b) × (s-c)) **Perimeter:** P = a + b + c **Law of Cosines (find angles):** cos(A) = (b² + c² - a²) / (2bc) cos(B) = (a² + c² - b²) / (2ac) cos(C) = (a² + b² - c²) / (2ab) Then angle = arccos(value). **Worked example: 3-4-5 right triangle** s = (3 + 4 + 5) / 2 = 6. A = √(6 × 3 × 2 × 1) = √36 = 6. Perimeter: 3 + 4 + 5 = 12. Angles: cos(A) = (16 + 25 - 9)/(2 × 4 × 5) = 32/40 = 0.8. A = arccos(0.8) ≈ 36.87°. cos(B) = (9 + 25 - 16)/(2 × 3 × 5) = 18/30 = 0.6. B = arccos(0.6) ≈ 53.13°. cos(C) = (9 + 16 - 25)/(2 × 3 × 4) = 0/24 = 0. C = arccos(0) = 90°. Verify: 36.87 + 53.13 + 90 = 180° ✓ **Worked example: equilateral triangle (s = 5)** A = √(7.5 × 2.5 × 2.5 × 2.5) = √(117.19) ≈ 10.83. Or formula: A = (√3/4) × s² = (√3/4) × 25 ≈ 10.83. ✓ All angles 60°. **Triangle inequality:** For valid triangle: each side < sum of other two. For a = 3, b = 4, c = 5: 3 < 4+5 ✓, 4 < 3+5 ✓, 5 < 3+4 ✓. Valid. For a = 3, b = 4, c = 9: 9 > 3+4 = 7. Invalid. No such triangle. **Triangle classifications:** By angles: - Acute: a² + b² > c² (and similar for others). - Right: a² + b² = c² (Pythagorean theorem). - Obtuse: a² + b² < c² (where c is longest). By sides: - Equilateral: a = b = c. All angles 60°. - Isosceles: at least two sides equal. Base angles equal. - Scalene: all sides different. **Common triangles:** | Triangle | Sides | Area | Angles | |---|---|---|---| | 3-4-5 right | 3, 4, 5 | 6 | 37°, 53°, 90° | | 5-12-13 right | 5, 12, 13 | 30 | 23°, 67°, 90° | | Equilateral (1) | 1, 1, 1 | √3/4 ≈ 0.43 | 60, 60, 60° | | Equilateral (10) | 10, 10, 10 | 43.30 | 60, 60, 60° | | Isosceles 5-5-6 | 5, 5, 6 | 12 | 73.7°, 73.7°, 53.13° | | Random scalene | 7, 8, 9 | 26.83 | 48°, 58°, 73° | **Pythagorean triples:** Integer right triangle sides: 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41, 20-21-29. Useful for clean-number geometry problems. **Centroid:** The centroid (geometric center) is the intersection of three medians (lines from vertex to opposite side midpoint). Coordinates: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). Centroid divides each median in 2:1 ratio. **Incircle (inscribed circle):** Largest circle inscribed in triangle. Radius: r = A / s (area / semi-perimeter). For 3-4-5: r = 6/6 = 1. Centered at incenter. **Circumcircle (circumscribed circle):** Smallest circle through all three vertices. Radius: R = (abc) / (4A). For 3-4-5: R = 60/24 = 2.5. Centered at circumcenter. **Triangle areas by other formulas:** - **Base and height**: A = (1/2) × base × height. - **Two sides and included angle**: A = (1/2) × a × b × sin(C). - **Coordinates** (Shoelace): A = (1/2) × |x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|. - **Vectors**: A = (1/2) × |v₁ × v₂| (cross product magnitude). **Triangle angle sum:** A + B + C = 180° (always). Find third angle from other two. **Special triangles:** **30-60-90 triangle:** Sides in ratio 1 : √3 : 2. For shortest side x: other sides x√3 and 2x. **45-45-90 (isosceles right):** Sides in ratio 1 : 1 : √2. Two legs equal, hypotenuse √2 times leg. **Triangle inequality applications:** - **Land plots**: ensure triangle is valid before computing. - **Truss design**: members must satisfy inequality. - **Navigation**: triangulating positions. - **Game design**: collision detection with triangles. **Common applications:** - **Surveying**: triangulating land. - **Navigation**: bearing-based position fixing. - **Engineering**: truss analysis. - **Architecture**: roof rafters, structural elements. - **Computer graphics**: 3D meshes are triangles. - **Astronomy**: stellar parallax. - **Photography**: triangulating focus. - **Robotics**: forward kinematics. **Numerical example: triangulation** Surveyor measures from two known points A and B (distance d = 200 m apart) to point C. Distances: AC = 150 m, BC = 180 m. To find triangle area: s = (200 + 150 + 180) / 2 = 265. A = √(265 × 65 × 115 × 85) ≈ √(168,251,875) ≈ 12,971 m². Hmm, but actually: Wait, this is a much smaller triangle. Let me redo: Actually for points where the distance from observation points are 150 and 180 with baseline 200, the triangle has these specific dimensions. A = √(265 × 65 × 115 × 85) ≈ √(168,251,875) ≈ 12,971 m² (~1.3 hectares). **Software:** - **CAD**: built-in triangle tools. - **Excel**: Heron's formula in cell. - **GIS**: triangle/polygon calculations. - **Programming**: simple math. - **Trigonometry calculators**: usually have triangle solvers. **Pitfalls:** - **Triangle inequality**: verify validity. - **Sides as angles**: don't confuse. - **Right vs obtuse**: check Pythagorean inequality. - **For very thin triangles**: numerical precision issues. - **Heron's formula precision**: subtraction near-equal can lose precision for thin triangles. **Educational notes:** Triangle properties form: - 5th-6th grade: classification. - 7th-9th grade: area, perimeter, Pythagorean. - High school: law of sines/cosines, trig. - College: vector treatment, computer graphics. Fundamental geometric shape; foundation for trigonometry. **Programming:** import math def heron_area(a, b, c): s = (a + b + c) / 2 return math.sqrt(s * (s-a) * (s-b) * (s-c)) def angle_from_sides(a, b, c): # angle opposite side a return math.degrees(math.acos((b**2 + c**2 - a**2) / (2*b*c))) Trivial in any language. **Pitfalls (continued):** - **For thin triangles**: small angles produce numerical issues. - **For very large triangles**: floating-point precision. - **For coordinate-based**: Shoelace formula works for any polygon. - **For 3D triangles**: use vector cross product method.