How this instrument works
The Triangle Sum Theorem states that the three interior angles of any triangle, on a flat plane, always add up to exactly 180° — a right angle and a half. This holds regardless of whether the triangle is tall and narrow, wide and flat, acute, obtuse, or a perfect equilateral triangle with three equal 60° angles; the total never changes. A quick proof: draw a line through one vertex parallel to the opposite side, and the alternate-interior angles formed exactly recreate the other two angles alongside the original, laid flat along a straight line, which is itself always 180°.
Because the total is fixed, knowing any two of a triangle's three angles is enough to recover the third with simple subtraction: the missing angle equals 180° minus the sum of the two known ones. This is one of the most-used facts in geometry, showing up anywhere a triangle's shape needs completing from partial angle information — surveying, truss design, and basic trigonometric problem-solving all lean on it constantly.
The theorem also sets a hard limit on what two angles can validly describe a triangle at all: their sum must stay strictly below 180°, since the third angle has to be a positive amount. Two angles that already sum to 180° or more cannot belong to any real triangle — there would be nothing left over for the third corner.
- Enter one known interior angle into the Angle A field.
- Enter the second known interior angle into the Angle B field.
- Read Angle C: the sheet subtracts both from 180° and reports the missing third angle.
- If your two angles already sum to 180° or more, the sheet will flag the input, since no triangle can have angles like that.
Worked example — two angles of 50° and 60°
A triangle has one angle measured at 50° and another at 60°. The third angle is C = 180° − 50° − 60° = 70° exactly — no measurement of the third corner was needed at all, since the theorem alone pins it down completely once the other two are known.
Compare a right triangle with one angle fixed at exactly 90° and a second angle of 45°: the third angle is 180° − 90° − 45° = 45°, making this the familiar 45-45-90 isosceles right triangle, where two of the three angles happen to match.
Questions
What is the Triangle Sum Theorem?
It states that the three interior angles of any triangle always add up to exactly 180°, regardless of the triangle's shape or size. It is one of the foundational facts of Euclidean (flat-plane) geometry, provable by drawing a line through one vertex parallel to the opposite side.
How do I find a missing triangle angle if I know the other two?
Subtract both known angles from 180°: missing angle = 180° − angle 1 − angle 2. Since the three angles always total exactly 180°, this subtraction is guaranteed to give the correct third angle with no further information needed.
Can two given angles ever fail to make a valid triangle?
Yes — if the two given angles already sum to 180° or more, no third positive angle can complete the triangle, since all three must add to exactly 180°. That combination describes a geometrically impossible triangle, and this calculator flags it rather than returning a negative angle.
Does the Triangle Sum Theorem work on a sphere or curved surface?
No — 180° is specific to triangles drawn on a flat (Euclidean) plane. A triangle drawn on a sphere's curved surface, such as one bounded by great-circle arcs on the globe, has interior angles that sum to MORE than 180°, with the excess growing as the triangle covers more of the sphere.
How is this different from the exterior angle theorem?
The Triangle Sum Theorem covers the three INTERIOR angles, which sum to 180°. The exterior angle theorem is a related but separate statement about a triangle's OUTSIDE angles: any exterior angle equals the sum of the two non-adjacent interior angles, a consequence of the same 180° total.