Calculate all three angles of a triangle from three side lengths using the law of cosines. Shows angle classification, radian conversion, sum verification, area, and full triangle properties.
The triangle angles calculator finds all three interior angles of a triangle when you know the three side lengths. It uses the law of cosines — the generalization of the Pythagorean theorem to any triangle — which states cos A = (b² + c² − a²) / (2bc). By applying this formula three times (once for each angle), the calculator recovers the complete angle set.
Knowing all three angles unlocks a wealth of information about a triangle. The calculator classifies the triangle by angle type: acute (all angles < 90°), right (one angle = 90°), or obtuse (one angle > 90°). It also classifies by side relationships: equilateral (all equal), isosceles (two equal), or scalene (all different). Every angle is displayed in both degrees and radians.
The tool also computes the area (using Heron's formula and the ½ab sin C formula), perimeter, semi-perimeter, all three altitudes, the circumradius R, and the inradius r. A verification row confirms that angles A + B + C = 180° exactly, giving you confidence in the results.
Presets cover the most commonly studied triangles — equilateral, isosceles, 3-4-5 right triangle, 5-12-13, and more. A reference table lists classic triangle families with their side ratios and angle values. Whether you are solving homework problems, checking survey measurements, or validating CAD designs, it gives a comprehensive solution.
This calculator is useful when the side lengths are known but the angles are not. It converts an SSS triangle into full angle data, classifies the shape, and cross-checks the geometry with area, radii, and altitude outputs, which makes it practical for homework verification, layout work, fabrication checks, and surveying problems where measured lengths need to be turned into reliable interior angles.
Law of cosines: cos A = (b² + c² − a²) / (2bc)
Angle A = arccos((b² + c² − a²) / (2bc))
Heron's formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2
Area (trig): ½ × a × b × sin C
Circumradius: R = (a × b × c) / (4 × Area)
Inradius: r = Area / s
Altitude to a: hₐ = 2 × Area / aResult: Angle A ≈ 36.87°, Angle B ≈ 53.13°, Angle C = 90°, perimeter = 12
For sides 3, 4, and 5, the law of cosines gives cos(A) = (4^2 + 5^2 − 3^2) / (2·4·5) = 0.8, so A ≈ 36.87°. Repeating the calculation for B gives about 53.13°, and the third angle is 180° minus their sum, which confirms a right triangle. The same side set also gives perimeter 12 and area 6, so the angle solve agrees with the rest of the triangle data.
When all three sides are known, the law of cosines is the standard way to recover the interior angles. A good workflow is to compute the angle opposite the longest side first, because that immediately tells you whether the triangle is acute, right, or obtuse and gives a stable starting point for the rest of the solve. After two angles are found, the third is best obtained from the 180 degree angle sum to keep rounding error under control.
The completed angle set does more than label the triangle. It tells you how the shape will behave in trigonometric area formulas, whether a longest side can act as a diameter in circle problems, and how steep or flat the triangle is when used in a structural or surveying layout. Equal angles reveal an isosceles relationship immediately, and three 60 degree angles confirm the equilateral case without any further side comparison.
Showing the same triangle in both degrees and radians is useful because geometry classes, CAD systems, and programming libraries do not all use the same angle unit. The degree output is easy to interpret visually, while the radian form is the one most often needed for formulas and code. A separate sum check also matters: if A + B + C is not 180 degrees within rounding, the side inputs or a hand calculation upstream are wrong.
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The law of cosines states c² = a² + b² − 2ab cos C. It generalizes the Pythagorean theorem (which is the special case when C = 90°) and lets you find any angle from three known sides.
They must satisfy the triangle inequality: every side must be less than the sum of the other two. Equivalently, the longest side must be less than the sum of the other two.
Degrees divide a full circle into 360 parts; radians use the radius as a unit, so a full circle is 2π radians. One radian ≈ 57.296°.
If all angles are less than 90°, it is acute. If one angle equals 90°, it is a right triangle. If one angle exceeds 90°, it is obtuse.
Yes, absolutely. Enter the three sides and the calculator will find the angles, including the 90° one. It also identifies the triangle as a right triangle automatically.
Heron's formula computes the area from three sides: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. No angles needed.
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Solve any right triangle from two known values — legs, hypotenuse, or angles. Computes all sides, angles, area, perimeter, inradius, circumradius, and altitude. Includes Pythagorean triple presets.