Triangle Calculator

This triangle calculator "solves" a triangle: given just enough known measurements, it works out every remaining side and angle, then reports the area, perimeter, and the incircle and circumcircle radii. It also classifies the triangle as acute, right or obtuse and as scalene, isosceles or equilateral. It is built for geometry and trigonometry students, surveyors and draughtspeople, carpenters and DIY builders, and anyone solving a triangle for coursework or a real layout. Pick the input mode that matches what you already know — SSS, SAS, ASA or SSA — and the tool applies the correct law automatically.

The four input modes

A triangle is fully determined by three pieces of information, as long as at least one of them is a side. SSS means all three sides are known. SAS means two sides and the angle between them (the included angle). ASA means two angles and the side between them. SSA means two sides and an angle opposite one of them — the famous "ambiguous case", which can yield zero, one or two valid triangles. (Three angles alone, AAA, fixes the shape but not the size, so it cannot be solved for lengths.)

The laws that do the work

Two rules cover every case. The Law of Cosines, c² = a² + b² − 2ab·cos(C), generalises Pythagoras: when the included angle C is 90°, cos(C) = 0 and it collapses to c² = a² + b². It is used to find the third side in SAS, and rearranged as cos(A) = (b² + c² − a²) / 2bc to recover an angle in SSS. The Law of Sines, a/sin(A) = b/sin(B) = c/sin(C), says each side is proportional to the sine of its opposite angle; it handles ASA and SSA. Underpinning both is the angle-sum rule: the three interior angles of any triangle always add to 180°, so once two angles are known the third is just 180° − A − B.

Worked example — SSS

Take sides a = 5, b = 7, c = 8. First check the triangle inequality (any two sides must exceed the third): 5 + 7 = 12 > 8, so it is valid. Find angle A opposite side a with the Law of Cosines: cos(A) = (7² + 8² − 5²) / (2 × 7 × 8) = (49 + 64 − 25) / 112 = 88 / 112 ≈ 0.7857, so A ≈ 38.21°. Likewise cos(B) = (5² + 8² − 7²) / (2 × 5 × 8) = 40 / 80 = 0.5, so B = 60°, and C = 180 − 38.21 − 60 ≈ 81.79°. The area comes from Area = ½ab·sin(C) = ½ × 5 × 7 × sin(81.79°) ≈ 17.32 square units. These are exactly the figures the calculator produces for the default SSS inputs.

Area, radii and classification

Once all sides and angles are known the tool computes area with ½ab·sin(C), perimeter as a + b + c, the incircle radius r = Area ÷ s where s is the semi-perimeter (perimeter ÷ 2), and the circumcircle radius R = abc ÷ (4 × Area). It then labels the triangle: right if any angle is 90°, obtuse if one exceeds 90°, otherwise acute; and equilateral, isosceles or scalene by comparing side lengths.

Tips and common mistakes

Match each side to the angle directly opposite it — side a faces angle A — or the laws give nonsense. In SAS the angle must be the one between the two given sides; in ASA the side must be the one between the two angles. Be alert to the SSA ambiguous case: when two answers are possible, this calculator shows the second triangle in an amber panel rather than hiding it. If your inputs break the triangle inequality, or two angles already sum to 180° or more, no triangle exists and the tool says so. Keep angles in degrees here (it converts to radians internally), and remember the area formula needs the sine of the included angle, not just any angle.