Exterior Angles of a Triangle Calculator

Calculate the exterior angles of any triangle from its interior angles or side lengths.

Enter Triangle Information

Result

Sum of Exterior Angles
360°
degrees
Interior Angle A60°
Exterior Angle A120°
Interior Angle B70°
Exterior Angle B110°
Interior Angle C50°
Exterior Angle C130°
Triangle TypeAcute

Step-by-Step Solution

Exterior Angle = 180° - Interior Angle

Understanding Exterior Angles of a Triangle

An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. Each vertex of a triangle has two exterior angles (one on each side), but they are always equal because they are vertical angles' supplements. The most important properties are: each exterior angle equals 180 degrees minus the interior angle, and the sum of all three exterior angles (one at each vertex) always equals 360 degrees.

Key Theorems and Properties

Exterior Angle Theorem

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Ext A = Interior B + Interior C

Supplementary Property

Each exterior angle and its adjacent interior angle are supplementary (sum to 180 degrees).

Ext Angle = 180° - Int Angle

Sum of Exterior Angles

The sum of the three exterior angles of any triangle is always 360 degrees.

Ext A + Ext B + Ext C = 360°

Law of Cosines

When given sides, use the Law of Cosines to find interior angles first.

cos A = (b²+c²-a²) / (2bc)

Equilateral Triangle

All interior angles are 60 degrees, so all exterior angles are 120 degrees.

Each exterior = 120°

Remote Interior Angles

The exterior angle is always greater than either of the two non-adjacent interior angles.

Ext A > Int B and Ext A > Int C

How to Calculate Exterior Angles

The simplest method is to subtract each interior angle from 180 degrees. If you know two interior angles, find the third using the fact that interior angles sum to 180 degrees, then compute all exterior angles. If you know the side lengths, use the Law of Cosines to find each interior angle, then subtract from 180 degrees.

Practical Applications

  • Navigation and surveying use exterior angles for direction changes at turning points.
  • Architecture and construction rely on precise angle calculations for structural integrity.
  • The sum of exterior angles property extends to all convex polygons (always 360 degrees).
  • Robotics and pathfinding algorithms use exterior angles to calculate turn angles.
  • Exterior angles help prove many geometric theorems and solve complex problems.