Every triangle's three angles add to exactly 180 degrees — no exceptions, on any flat surface anywhere. That single fact solves half of all angle problems: know two angles and the third is 180 minus their sum. A roof truss with two 35° base angles leaves a 110° peak, and the carpenter cuts the lumber to match. When sides enter the picture, the law of sines takes over: each side divided by the sine of its opposite angle gives the same constant, so a known side can unlock an unknown angle through a product ratio.

Surveyors laying out property lines and navigators triangulating positions use these relationships daily. This calculator handles the product step inside the law of sines: type the known side length in the Variable A box and the sine ratio (sin unknown / sin known) in the Variable B box, and the Result box gives the unknown side — or rearrange the same products to solve for a missing angle. A 100-foot lot line opposite 50° with a second angle of 60° yields 100 × (sin 60°/sin 50°) ≈ 113 feet of frontage.

The angle-sum method, worked three ways

The angle sum is the fastest tool in triangle work, and it deserves three clean examples. A roof truss has two base angles of 38 degrees each, so the peak angle is 180 − 38 − 38 = 104 degrees, and the ridge cut is made to match. A right triangle has one 90-degree angle and one 27-degree angle, so the third angle is 180 − 90 − 27 = 63 degrees. An isosceles triangle has a vertex angle of 46 degrees, so the two equal base angles share the remaining 134 degrees equally: (180 − 46) / 2 = 67 degrees each. In every case the check is the same: add the three angles back up and confirm the total is exactly 180. A sum of 179 or 181 means a measurement or an arithmetic slip, not a new kind of triangle.

Naming triangles by their angles

Angles also classify triangles, and the classification tells you what to expect before any calculation. An acute triangle has all three angles below 90 degrees, like the 60-60-60 equilateral triangle or a 52-71-57 triangle. A right triangle has exactly one 90-degree angle, and its other two angles always add to 90 degrees. An obtuse triangle has exactly one angle above 90 degrees, like the 104-degree truss peak above. Two angles above 90 degrees are impossible in one triangle, because they would already exceed 180 degrees before the third angle is counted, and two right angles fail for the same reason: 90 plus 90 already spends the whole budget. If a problem hands you two obtuse angles, the measurements are wrong, not the theorem.

The exterior angle shortcut

Every triangle has an exterior angle at each corner, formed by extending one side past the vertex, and it equals the sum of the two remote interior angles. A corner with an interior angle of 65 degrees has an exterior angle of 180 − 65 = 115 degrees, and the same 115 degrees comes from adding the two remote interiors, 40 + 75 = 115. Roofers use this constantly: the angle a rafter makes with the horizontal outside the triangle is the exterior angle, and it can be read off the two far corners without measuring at the joint itself. The exterior shortcut also double-checks angle-sum work, since both routes must agree.

The law of sines, with a second worked example

The lot-line example above shows the law of sines in its product form: side over sine of its opposite angle is constant across all three sides. A second example, with different numbers, locks the method in. A surveyor knows one boundary of 85 feet opposite a 42-degree angle and needs the boundary opposite a 68-degree angle. The sine ratio is sin 68° / sin 42° = 0.9272 / 0.6691 ≈ 1.3857, and the unknown side is 85 × 1.3857 ≈ 117.78 feet. On this calculator, type 85 in the Variable A box and 1.3857 in the Variable B box, and the Result box shows about 117.78. The calculator performs the multiplication; looking up the sines and forming the ratio is the by-hand step before it. Keep the calculator in degree mode for the sine lookup, because a radian-mode sine of 68 gives a completely different ratio and a silently wrong side.

The law of cosines for angle hunting

When all three sides are known but no angle is, the law of cosines finds the angles: for sides a, b, c with angle C opposite side c, cos C = (a² + b² − c²) / (2ab). Take a triangle with sides 7, 8, and 9 units. The angle opposite the 9 side has cos C = (49 + 64 − 81) / (2 × 7 × 8) = 32 / 112 ≈ 0.2857, so C ≈ 73.40 degrees. The angle opposite the 7 side has cos A = (64 + 81 − 49) / (2 × 8 × 9) = 96 / 144 ≈ 0.6667, so A ≈ 48.19 degrees. The third angle needs no new formula: 180 − 73.40 − 48.19 = 58.41 degrees, and the three sum to 180 as they must. The law of cosines also settles whether a triangle is acute, right, or obtuse at a glance: if a² + b² exceeds c² for the longest side, the triangle is acute; if it equals c², it is right; if it falls short, it is obtuse.

The ambiguous case, and why SSA needs a second look

Two sides and a non-included angle, the SSA configuration, can describe two different triangles, and missing the second one is the classic law-of-sines error. Suppose a side of 10 units sits opposite a 30-degree angle, and another side of 14 units is known. Then sin B = 14 × sin 30° / 10 = 14 × 0.5 / 10 = 0.7, and the inverse sine gives B ≈ 44.43 degrees. But 180 − 44.43 = 135.57 degrees has the same sine, and 30 + 135.57 = 165.57 is still under 180, so a second valid triangle exists with B ≈ 135.57 degrees. Always test the supplement: if the given angle plus the supplementary angle stays under 180 degrees, report both triangles. Surveyors who skip this test stake out the wrong lot line half the time on SSA jobs.

Degrees, radians, and the mode trap

Angles in school and construction are measured in degrees, but calculators and code often default to radians, where a full circle is 2π instead of 360 degrees. The conversion is 180 degrees = π radians, so 45 degrees = 0.7854 radians and 90 degrees = 1.5708 radians. The single most common triangle-angle error is taking the sine of a degree number in radian mode: sin(68) in radians is about 0.411, not 0.9272, and every side computed from it is wrong by more than half. Before any sine or inverse-sine step, confirm the mode shows DEG, and after any answer, confirm the angles still sum to 180.

Five mistakes that break triangle angles

The first mistake is forgetting the angle sum, leaving a third angle uncomputed or inconsistent. The second is radian-mode sines, which corrupt every law-of-sines product silently. The third is ignoring the ambiguous case in SSA problems and reporting one triangle where two exist. The fourth is assuming a triangle is right because it looks right in a sketch; only a 90-degree angle or a 3-4-5-style side check earns that label. The fifth is mixing up opposite pairs in the law of sines, pairing a side with an angle that is not across from it, which produces a ratio that looks plausible and means nothing. One habit catches all five: after solving, verify the angle sum, verify each side sits opposite its own angle, and verify the longest side sits opposite the largest angle.

Related calculators on this site

If the angles are part of a bigger job, companion tools carry the next step. The triangle area calculator turns known sides and angles into area, including the two-sides-and-included-angle route. The right triangle calculator handles the 90-degree special case, where one angle is already known and the Pythagorean theorem replaces the law of cosines. The slope calculator converts a rise-over-run into an angle, the exact step for ramps and roof pitches. The trigonometry calculator evaluates the individual sines, cosines, and tangents these methods need. Each one uses the same discipline: name the angle, confirm the mode, and check the sum.

A classroom note on why the sum is 180

Students often ask why the angles must add to 180 degrees rather than some other number. Draw any triangle, then draw a line through one vertex parallel to the opposite side. The two base angles reappear as alternate interior angles along that line, sitting beside the vertex angle, and the three angles together form a straight line, which is 180 degrees by definition. The proof needs only parallel lines and a straight angle, which is why the sum holds for every flat triangle and fails only on curved surfaces, where surveyors working over whole continents must correct for it.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Triangle Angles Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.