Triangle Degree Calculator - Solve Any Triangle

A triangle degree calculator is a tool that returns the three interior angles of a triangle in degrees for whichever combination of values you already have: three side lengths, two angles, or two sides plus the angle between them.

• • Geometry homework and exam problems: Solve a problem that hands you three sides, two angles, or two sides with the included angle, and read off alpha, beta, gamma without re-deriving the law of cosines or sines.
• • Roof pitch and rafter cuts: Convert a measured run, rise, and rafter length into the pitch angles so a carpenter can set the saw to the right degrees and the framing square to the right number.
• • Surveying and slope work: Turn a slope distance and a measured angle into a horizontal leg and a vertical rise, which is what slope percentage, drainage fall, and ramp grades are based on.

The label degree in the name refers to the unit of the output. Every interior angle is reported in degrees, the same unit that protractors, bevel squares, and most classroom problems use.

For a triangle solver that leans on the same alpha, beta, gamma notation and the law of cosines, the Triangle Angle Calculator is the closest peer in the same Math & Conversion cluster.

The calculator reads which combination of values is filled in, then routes to the matching rule. Three sides use the law of cosines, two angles use the 180 degree sum, two sides and the included angle use the law of cosines followed by the law of sines, and two sides plus a non-included angle use the law of sines. Invalid inputs are caught before any number is shown.

• a, b, c: the three side lengths of the triangle, each opposite the matching interior angle, all in the same length unit
• alpha, beta, gamma: the three interior angles in degrees, each opposite the matching side, and they always sum to 180
• s: the semi-perimeter (a + b + c) / 2, used by Heron's formula for the area

When the input is three sides, the law of cosines gives each angle from the two sides that touch it. When the input is two angles, the third angle falls out of the 180 degree sum.

According to Wolfram MathWorld, the law of cosines gives the angle opposite side a as alpha = arccos((b^2 + c^2 - a^2) / (2bc)) for a triangle with sides a, b, c

To see the same cosine-based angle rule worked out for any pair of sides, the Law of Cosines Calculator is the standalone reference that the three-side path of this calculator is built on.

These four ideas cover every rule the calculator uses to recover the missing angles and the missing side.

With these four facts, every supported input combination resolves to a unique triangle, and impossible combinations are rejected before any angle is shown.

When the input is two sides and a non-included angle, the sine ratio from the Law of Sines Calculator is the rule that closes the triangle without re-deriving the side-to-angle relationship.

Pick the input combination that matches the numbers you already have.

1. 1 Pick a single length unit: Use meters, feet, inches, or any one unit for all three side inputs. Mixing units is the most common cause of a wrong angle.
2. 2 Choose 3 sides if you have all three side lengths: Pick '3 sides (a, b, c)' from the dropdown and enter a, b, c. The law of cosines returns alpha, beta, gamma, the area, and a 180 degree sum check.
3. 3 Choose 2 angles if you have two interior angles: Pick '2 angles' and enter the two known angles. The third angle is 180 minus their sum.
4. 4 Choose SAS if you have two sides and the angle between them: Pick '2 sides + included angle (SAS)' and enter the two sides plus the angle at the corner where they meet. The third side comes from the law of cosines, then the two missing angles fall out of the law of sines.
5. 5 Read the angle sum check: The angle sum should read 180.00 for any valid triangle. A small mismatch means the inputs were inconsistent.

Suppose a roof rafter sits 4 m up the wall and runs 3 m out to the wall plate. Pick '2 sides + included angle (SAS)', enter a = 4, b = 3, and gamma = 53.13 degrees. The rafter length comes out as 5 m, the wall angle alpha is 36.87 degrees, the rafter angle beta is 90 degrees, and the area is 6 square meters, matching the 3-4-5 right triangle in degrees.

If the same sides are also feeding a coverage or material estimate, the Triangle Area Calculator takes the three side lengths and returns the area with the same Heron's formula path.

The triangle degree calculator covers every supported input combination in one tool and reports the result with the same three-angle plus area layout each time.

• • Handles all four supported input combinations: Three sides, two angles, two sides + included angle, and two sides + non-included angle are all solved from the same form.
• • Shows alpha, beta, and gamma together: All three interior angles land in the same result panel, so a follow-up step such as a bevel cut, a slope percentage, or a bearing calculation can use the value it needs without re-entering data.
• • Reports the 180 degree angle sum check: The result panel prints the sum of the three angles. A value other than 180 flags an inconsistent input set, which is what users typically miss when a hand calculation drifts by a degree or two.
• • Catches impossible triangles before showing angles: The triangle inequality is enforced on three-side inputs, the angle sum is enforced on two-angle inputs, and the ambiguous SSA case is rejected when sin of the unknown angle would have to be greater than 1.

The area is computed from Heron's formula once the three sides are known, so a roofing, flooring, or sheathing estimate can be pulled from the same result.

For the broader triangle page that accepts base-height, Heron, and SAS inputs in one place, the Triangle Calculator is the natural follow-up when a problem is not framed in pure degrees.

A few input choices and assumptions decide whether the angle result matches the actual triangle you are trying to solve.

• • Two angles alone do not fix the size of the triangle. The 180 degree sum gives the third angle, but the area and the side lengths stay unknown until at least one side is added.
• • The result is the geometric solution of an ideal triangle. Real construction or surveying work usually needs extra allowance for material thickness, slope, or boundary clearance, which the calculator does not add on its own.

If your sides come from a drawing, double-check whether the values are edge-to-edge or include a wall, fence, or material thickness, and strip that thickness before entering the sides.

According to Wolfram MathWorld, the law of sines states that for a triangle with sides a, b, c and opposite angles alpha, beta, gamma, a / sin(alpha) = b / sin(beta) = c / sin(gamma)

According to Math Open Reference, the three interior angles of a Euclidean triangle always add up to 180 degrees, so the third angle is 180 minus the sum of the other two

If a downstream tool needs the angles in radians, gradians, or turns instead of degrees, the Angle Converter handles the unit shift without redoing the triangle math.