Triangle Angle Calculator - Alpha, Beta, Gamma from Sides or Angles

The triangle angle calculator solves for the three interior angles alpha, beta, and gamma of a triangle from whatever mix of sides and angles you already know. It accepts three side lengths, two sides and an angle, or two angles, and returns the missing pieces using the law of cosines, the law of sines, and the angle sum theorem.

• • Geometry homework and worksheet checks: Recover alpha, beta, gamma from three side lengths, two sides plus an included angle, or a non-included side-angle pair.
• • Roof, rafter, and stair stringer cuts: Convert a measured run, rise, and pitch angle into the cut angle and the rafter length without a construction calculator.
• • Surveying and slope work: Turn a measured slope distance and a horizontal angle into the third side and the two missing interior angles.
• • Navigation and bearing problems: Solve a triangle on a chart from two bearings and the distance between them, and read out the third bearing as an interior angle.

Alpha, beta, and gamma are the three interior angles, sitting opposite sides a, b, and c. The law of cosines is the workhorse for three sides or two sides and the angle between them. The law of sines takes over for two sides and a non-included angle. The angle sum theorem is the safety net: once two angles are known, the third is 180 minus the other two.

When the recovered alpha is exactly 90 degrees, the Right Triangle Calculator is the closest peer because the same Pythagorean-style label applies.

The calculator reads which of the six values you supplied, picks the matching law, and reports alpha, beta, gamma together with a 180 degree sum check.

• a, b, c: The three side lengths, in the same length unit, opposite alpha, beta, and gamma
• alpha, beta, gamma: The three interior angles in degrees, each strictly between 0 and 180
• 180 degree sum: The consistency check that confirms alpha + beta + gamma

When three sides are known, the law of cosines is the cleanest path: each angle is the arccosine of a ratio built from the other two sides. The triangle inequality check runs first so an impossible combination is rejected before any arccosine is taken. When two sides and an angle are known, the calculator picks the law of cosines for SAS or the law of sines for SSA, and the angle sum theorem closes the loop.

According to Wolfram MathWorld, the law of cosines states that for a triangle with sides a, b, c opposite angles alpha, beta, gamma, c^2 = a^2 + b^2 - 2ab cos(gamma), and the law can be rearranged to recover an angle as gamma = arccos((a^2 + b^2 - c^2) / (2ab)).

According to Wolfram MathWorld, the law of sines states that for any triangle, a / sin(alpha) = b / sin(beta) = c / sin(gamma), so a known side and its opposite angle can be paired with a second side to recover its opposite angle as beta = arcsin(b * sin(alpha) / a).

For a deeper derivation of the cosine rule that powers the three-side case, the Law of Cosines Calculator walks through the algebra step by step.

These four ideas are the toolkit the calculator uses, and they are the rules taught in any first geometry or trigonometry course.

These four ideas cover every input combination the calculator accepts. The law of cosines is preferred for SSS and SAS, the law of sines for SSA, and the angle sum theorem closes every case. The triangle inequality is the gatekeeper that filters bad inputs.

To see the SSA branch worked out with the ambiguous case treated explicitly, the Law of Sines Calculator covers that side of the problem.

Pick the input combination that matches the data you have, fill in the matching fields, and read alpha, beta, gamma plus the missing side from the result panel.

1. 1 Pick a single length unit for sides a, b, c: Use meters, feet, or inches for every side input. Mixing units skews the recovered angles.
2. 2 Enter three sides for a full SSS problem: Type side a, side b, and side c, and leave alpha, beta, gamma blank. The law of cosines recovers all three angles and the 180 degree sum check.
3. 3 Enter two sides and the included angle for SAS: Fill in the two sides that meet at the known angle and the angle between them. The law of cosines recovers the missing side and the angle sum fixes the remaining angles.
4. 4 Enter two sides and a non-included angle for SSA: Fill in the two sides and the angle opposite one of them. The law of sines recovers the second angle, and the third side follows.
5. 5 Enter two angles to recover the third: Fill in any two of alpha, beta, gamma and leave the sides blank. The third angle is 180 minus the other two, but sides cannot be recovered without a length input.
6. 6 Read the result panel and the 180 degree sum check: Use the recovered angles for cut lists, bearing work, or worksheet answers, and watch the sum field for any drift that would flag an inconsistent measurement.

A roof rafter sits against a 4 m wall and runs 3 m out from the base, with the wall angle at 90 degrees. Enter side a = 4, side b = 3, and angle beta = 90. The calculator returns the rafter length as 5 m and the remaining angles as 36.87 and 53.13 degrees.

When the inputs are three side lengths with no angles at all, the SSS Triangle Calculator focuses on the SSS case and returns the matching area, perimeter, and angle set.

The page accepts every realistic input mix a geometry problem can hand you, replacing a stack of separate SSS, SAS, ASA, and SSA solvers with a single result.

• • Covers every common input combination: Three sides, two sides plus an included angle, two sides plus a non-included angle, and two angles all return alpha, beta, gamma without switching tools.
• • Returns a 180 degree sum check: The result panel always shows alpha plus beta plus gamma so inconsistent measurements stand out immediately.
• • Matches law of cosines and sines notation: Side a sits opposite alpha, side b opposite beta, side c opposite gamma, the same labeling textbooks and engineering notes use.
• • Recovers the missing side for SAS and SSA: When only two sides are known, the calculator finishes the side trio so the result also feeds triangle area, perimeter, and law-of-cosines follow-ups.

For deeper law-of-cosines derivations, the law-of-cosines page goes one level further; for law-of-sines cases that need an obtuse alternative considered, the law-of-sines page handles that.

For the two-sides-plus-included-angle branch that recovers the third side first, the SAS Triangle Calculator is the peer that does the same work in SAS notation.

A few input choices decide whether the recovered angles match the actual triangle you are trying to solve.

• • Two angles alone cannot fix the size of the triangle. The calculator recovers the third angle from the angle sum theorem but returns no side length, because the law of sines still needs a length to scale the triangle.
• • The SSA case can have two valid solutions when the given side is shorter than the other known side. The calculator picks the acute solution and flags the ambiguity.
• • The result is the geometric solution of an ideal triangle. Real construction, surveying, or navigation work usually needs extra allowance for material thickness, slope, clearance, or measurement rounding.

These caveats match the warnings in the law of sines and law of cosines references. When the inputs are clean, the recovered sum stays within a hundredth of a degree of 180.

According to Wikipedia, the interior angles of any Euclidean triangle always sum to 180 degrees, so given two angles alpha and beta, the third angle is gamma = 180 - alpha - beta.

When the recovered angles need to feed a follow-up area or perimeter calculation, the Triangle Calculator accepts the full side set and returns the rest.