Calculate exterior angles of a triangle from interior angles or side lengths. Verify the exterior angle theorem, see remote interior angle pairs, and compare all angles visually.
Every triangle has three interior angles that sum to 180°. At each vertex, extending one side of the triangle beyond the vertex creates an exterior angle — the supplement of the interior angle at that vertex. The exterior angle theorem is one of the most important results in elementary geometry: each exterior angle equals the sum of the two non-adjacent (remote) interior angles.
For example, in a triangle with interior angles 50°, 60°, and 70°, the exterior angle at the 50° vertex is 130° — which equals 60° + 70°, confirming the theorem. The sum of one exterior angle at each vertex is always 360° for any convex polygon, and for a triangle this is easy to verify since the three exterior angles are (180 − A) + (180 − B) + (180 − C) = 540 − 180 = 360°.
If you know the side lengths instead of angles, the law of cosines lets you recover all interior angles: A = arccos((b² + c² − a²) / (2bc)), and similarly for B and C. From there, the exterior angles follow immediately.
This calculator supports two input modes — entering three interior angles directly, or entering three side lengths and computing angles via the law of cosines. It shows all interior and exterior angles, verifies the exterior angle theorem for each vertex, classifies the triangle, and displays a comparison bar chart plus a reference table of related theorems. Eight presets cover common triangles such as equilateral, right (45-45-90 and 30-60-90), and Pythagorean triples.
The Exterior Angles of a Triangle Calculator is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Exterior Angle at A, Exterior Angle at B, Exterior Angle at C in one pass, with conversions and derived values shown together.
Exterior angle at vertex A: ext_A = 180° − A
Exterior angle theorem: ext_A = B + C
Sum of exterior angles: ext_A + ext_B + ext_C = 360°
From sides (law of cosines): A = arccos((b²+c²−a²)/(2bc))Result: Exterior angles: 130°, 120°, 110°; sum = 360°
ext_A = 180−50 = 130° (also B+C = 60+70 = 130° ✓). ext_B = 180−60 = 120° (A+C = 50+70 = 120° ✓). ext_C = 180−70 = 110° (A+B = 50+60 = 110° ✓). Sum: 130+120+110 = 360°.
Exterior Angles of a Triangle Calculator calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
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An exterior angle is formed by one side of the triangle and the extension of an adjacent side. It is the supplement of the interior angle at that vertex: exterior = 180° − interior.
The exterior angle theorem states that an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. For example, if the interior angles are A, B, C, then the exterior angle at A equals B + C.
Each exterior angle is 180° − (interior angle). Sum = 3(180°) − (A+B+C) = 540° − 180° = 360°. This holds for any convex polygon: the sum of one exterior angle per vertex is 360°.
Yes, if the corresponding interior angle is greater than 90° (obtuse). For example, an interior angle of 120° gives an exterior angle of 60°.
Use the law of cosines: A = arccos((b²+c²−a²)/(2bc)), then B = arccos((a²+c²−b²)/(2ac)), and C = 180°−A−B.
The two interior angles that are NOT adjacent to a given exterior angle. For the exterior angle at vertex A, the remote interior angles are B and C.
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