Solve any triangle from three sides. Get the area, all three angles and the perimeter. Useful for geometry homework, land surveying and any structural geometry problem.
Area
6
Heron's formula
Perimeter
12
Angle A (opp. a)
36.8699°
Angle B (opp. b)
53.1301°
Angle C (opp. c)
90°
Sum of angles: 180° (always 180°)
لا يزال بعض المحتوى معروضًا بالإنجليزية.
Triangles are the most fundamental polygon in geometry - every other polygon can be decomposed into triangles, and computer graphics, structural engineering, surveying and trigonometry all rest on triangle calculations. This calculator solves the 'SSS' case (three sides given) using two classical formulas: Heron's formula for area (requires no angle or height) and the law of cosines for the interior angles. Together they give you a complete solution: area, all three angles, and a verification that the sides actually form a valid triangle.
For Class 9-10 students learning Heron's formula, this calculator is an instant verifier of homework. For Class 11-12 trigonometry students, it demonstrates the law of cosines in action with real numbers. For land surveyors, architects and structural engineers, it computes triangle areas from field measurements where heights are impractical to measure directly. The formula display lets you see exactly how the result was derived, supporting both verification and learning.
Enter the three side lengths.
Read area (via Heron's formula), interior angles (law of cosines), and perimeter.
Heron's formula computes area from the three sides without needing height. The law of cosines finds angles given sides: cos(A) = (b² + c² − a²) / (2bc). Sides must satisfy the triangle inequality (a + b > c) for a valid triangle.
s = (a + b + c) / 2 Area = √(s(s−a)(s−b)(s−c)) cos(A) = (b² + c² − a²) / (2bc)
Triangle inequality: for any valid triangle, the sum of any two sides must exceed the third. If a + b ≤ c (with c the longest side), no triangle exists - the points would be collinear or unreachable. The calculator checks this first and flags invalid inputs. Otherwise, it proceeds with Heron's formula and the law of cosines.
Heron's formula computes area from the three sides via the semi-perimeter s = (a+b+c)/2. Area = √(s(s−a)(s−b)(s−c)). The formula is over 2000 years old (attributed to Heron of Alexandria) and works for any triangle. Numerical stability tip: for very thin triangles (where one angle is near 0° or 180°), the formula can lose precision; this calculator uses the standard form which is fine for typical inputs but may need rearrangement for extreme cases.
Law of cosines: cos(A) = (b² + c² − a²) / (2bc) - and similarly for B and C. This gives the angle opposite each side. Always verify by checking that A + B + C ≈ 180° (small rounding error is normal). If the angles don't sum to 180°, your sides probably can't form a triangle, or there's a precision issue with the inputs.
Semi-perimeter: 6. Area = √(6×3×2×1) = √36 = 6. Angles: cos(A) = (16+25-9)/(2×4×5) = 0.8 → A = 36.87°. cos(B) = (9+25-16)/(2×3×5) = 0.6 → B = 53.13°. cos(C) = 0 → C = 90°. Sum: 180°. Confirms the classic 3-4-5 right triangle.
Semi-perimeter: 15. Area = √(15×5×5×5) = √1875 ≈ 43.30. All three angles = 60°. The expected result for an equilateral triangle.
Semi-perimeter: 90. Area = √(90×40×30×20) = √2,160,000 ≈ 1469.7 m². Useful for converting GPS-measured boundary distances to actual plot area for revenue records or registration.
The triangle inequality requires the sum of any two sides be greater than the third. Otherwise it's degenerate (collinear points). The calculator returns NaN if violated.
It needs only the three sides - no angle, no height. Other area formulas need at least one angle or the height.
Always - for any triangle in flat (Euclidean) geometry. If your computed angles don't sum to 180, your sides probably can't form a triangle.
Heron's formula loses precision for very thin / nearly degenerate triangles. The numerical-stable version sorts sides first; this calculator uses the standard form which is fine for typical inputs.
Use the law of sines: a/sin(A) = b/sin(B) = c/sin(C). This calculator focuses on the SSS case (three sides given).
Equilateral: all sides equal (60° everywhere). Isosceles: two sides equal. Scalene: all different. The calculator works for any of these.
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