Interior and Exterior Triangle Angles Calculator

Enter any two interior angles of a triangle to find the missing interior angle and all three exterior angles, using A + B + C = 180° and exterior = 180° − interior.

Quick Facts

Interior angle sum
A + B + C = 180°
True for every triangle, regardless of shape or size.
Exterior angle rule
Exterior = 180° − adjacent interior
Each exterior angle is supplementary to the interior angle at that vertex.
Exterior angle theorem
Exterior = sum of the two remote interior angles
The exterior angle at a vertex equals the sum of the two non-adjacent interior angles.
Exterior angle sum
Sum of all 3 exterior angles = 360°
Holds for every triangle, one exterior angle chosen per vertex.

Your Results

Calculated
Interior Angle C
-
180° − A − B
Exterior Angle A
-
180° − A
Exterior Angle B
-
180° − B
Exterior Angle C
-
180° − C

Ready

Enter two interior angles and choose a unit, then press Calculate.

Formula and method for Interior and Exterior Triangle Angles

Every triangle has three interior angles (the angles inside the triangle at each vertex) and three exterior angles (formed by extending one side outward at each vertex). The two are linked by two well-established Euclidean geometry facts: the interior angles always sum to 180°, and each exterior angle is supplementary to — that is, adds up to 180° with — the interior angle at the same vertex. This calculator uses those two rules together to find every angle in the triangle from just two known interior angles.

How the calculation works

Start from the triangle angle sum theorem: A + B + C = 180°, where A, B, and C are the three interior angles. If you know A and B, the missing interior angle is C = 180° − A − B. From there, each exterior angle is the supplement of its adjacent interior angle: exteriorA = 180° − A, exteriorB = 180° − B, and exteriorC = 180° − C. As a built-in check, the exterior angle theorem says each exterior angle also equals the sum of the two remote (non-adjacent) interior angles — for example exteriorA = B + C — and the three exterior angles always add up to exactly 360°.

Common mistakes

  • Entering an impossible pair: the two angles you enter must sum to less than 180° (or less than π radians). If A + B ≥ 180°, there is no valid triangle and the third angle would be zero or negative.
  • Mixing degrees and radians: pick one unit and enter both angles in it. A value of "60" means 60° in degree mode but 60 radians (an impossible angle) in radian mode.
  • Confusing exterior angle with the "outside" reflex angle: the standard exterior angle is the supplement of the interior angle (180° − interior), not 360° minus the interior angle.

Real-world applications

  • Carpentry and framing use interior and exterior triangle angles to cut miters and check that braces and roof trusses meet at the correct angle.
  • Navigation and surveying use triangle angle relationships to compute bearings and unknown angles from measured ones.
  • CAD, CNC, and 3D modeling rely on the angle sum and exterior angle rules to validate that polygon meshes and triangulated surfaces are geometrically consistent.
  • Geometry and trigonometry instruction use this relationship as a foundation for the exterior angle theorem and later work with polygon angle sums.

Frequently Asked Questions

What is the relationship between a triangle's interior and exterior angles?
Each exterior angle is supplementary to the interior angle at that same vertex, meaning they add up to 180°: exterior = 180° − interior. For example, an interior angle of 60° has an adjacent exterior angle of 120°.
What do the interior angles of a triangle add up to?
The three interior angles of any triangle always sum to 180° (the triangle angle sum theorem). If you know two angles, the third equals 180° minus the sum of the other two.
What do the exterior angles of a triangle add up to?
The three exterior angles of a triangle (one per vertex) always sum to 360°, the same total as any convex polygon's exterior angles.
What is the exterior angle theorem?
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 at B and C are 70° and 50°, the exterior angle at A is 70° + 50° = 120°, which matches 180° − 60°.

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