Triangle Sum Theorem Calculator

Background.

Cut the three corners off a paper triangle and lay them side by side, points together, and they make a straight line. That is the triangle sum theorem in one demonstration: the interior angles of any flat triangle add up to 180°, whatever its shape or size. A long thin sliver and a perfectly equilateral triangle obey the same rule, which is why knowing two angles always tells you the third.

The proof is short enough to remember. Draw a line through one corner parallel to the opposite side. The two angles that line makes with the triangle's other sides are equal to the triangle's other two angles, because alternate angles across parallel lines are equal, and together with the corner's own angle they fill a straight line. Euclid proved it with a similar parallel-line construction in Book I of the Elements, and the proof rests on his parallel postulate.

That dependence is not a technicality. On a curved surface the rule breaks. A triangle drawn on a globe from the North Pole down to the equator, a quarter of the way round the equator, and back up to the pole has three right angles, 270° in all. On surfaces that curve the other way, like a saddle, the angles add to less than 180°. For anything drawn on paper, cut from timber or set out on a building site, though, the flat-surface rule holds to far better accuracy than any protractor can measure.

Here you enter the two angles you know, in degrees, and get the third as 180° minus their sum. If the two angles already reach 180°, no triangle can close, and the page says so rather than returning zero or a negative angle.

What is triangle sum theorem calculator?

The triangle sum theorem states that the three interior angles of a triangle in a flat, Euclidean plane add up to exactly 180°, or π radians. An interior angle is the angle inside the triangle at one of its corners, between the two sides that meet there. Because the total is fixed, any one angle equals 180° minus the other two, and an angle at one corner can only grow if the other two shrink by the same amount between them.

How to use this calculator.

1. 01Enter the two angles you know, in degrees. If you have them in radians, multiply by 180 and divide by π first.
2. 02Make sure both are interior angles. If you measured an exterior angle, the one outside the triangle between a side and the extension of its neighbour, subtract it from 180° before entering it.
3. 03Check that the two angles add up to less than 180°; if they do not, the sides cannot meet and there is no triangle to complete.
4. 04Read the third angle. If it is exactly 90° the triangle is right-angled, above 90° it is obtuse, and if all three angles are under 90° it is acute.

The formula.

C = 180° − A − B, where A and B are the two known interior angles and C is the third. The 180° is the angle of a straight line, which is exactly what the three corners of a flat triangle fill when placed side by side. Nothing else enters the calculation: the lengths of the sides do not matter, so a triangle and any enlarged or shrunk copy of it have the same angles. The same fact gives the exterior angle theorem. The exterior angle at C, between one side and the extension of the other, is 180° − C, which equals A + B. Every degree added to A or B comes straight off C, so the answer is only as precise as the two angles you start with. The rule holds only on a flat surface; on a sphere the sum exceeds 180° by an amount that grows with the triangle's area.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.