Interior and Exterior Triangle Angles Calculator

Calculate all interior and exterior angles of a triangle. The sum of interior angles always equals 180 degrees.

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Result

Missing Angle C
75
degrees
Interior Angle A60\u00B0
Interior Angle B45\u00B0
Interior Angle C75\u00B0
Exterior Angle A120\u00B0
Exterior Angle B135\u00B0
Exterior Angle C105\u00B0
Triangle TypeAcute

Step-by-Step Solution

A + B + C = 180\u00B0 => 60 + 45 + C = 180 => C = 75\u00B0

Interior and Exterior Angles of a Triangle

One of the most fundamental properties of a triangle is that the sum of its three interior angles always equals 180 degrees (or pi radians). This is known as the Triangle Angle Sum Theorem and holds for all triangles in Euclidean geometry, regardless of their shape or size.

An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. Each exterior angle equals 180 degrees minus its adjacent interior angle. The Exterior Angle Theorem states that an exterior angle equals the sum of the two non-adjacent interior angles.

Key Formulas

Triangle Angle Sum

The three interior angles of any triangle always add up to 180 degrees.

A + B + C = 180\u00B0

Exterior Angle

Each exterior angle is the supplement of its adjacent interior angle.

Exterior = 180\u00B0 - Interior

Exterior Angle Theorem

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

Ext A = B + C

Sum of Exterior Angles

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

Ext A + Ext B + Ext C = 360\u00B0

Types of Triangles by Angles

  • Acute Triangle: All three interior angles are less than 90 degrees.
  • Right Triangle: One interior angle is exactly 90 degrees.
  • Obtuse Triangle: One interior angle is greater than 90 degrees.
  • Equilateral Triangle: All three angles are equal (60 degrees each).
  • Isosceles Triangle: Two angles are equal.
  • Scalene Triangle: All three angles are different.

Practical Applications

Understanding triangle angles is essential in construction, surveying, navigation, and engineering. Builders use the angle sum property to ensure structural integrity. Surveyors use it in triangulation to measure distances. Computer graphics rely on triangle angle calculations for rendering 3D models.

Tips for Solving Triangle Angle Problems

  • If you know two angles, the third is simply 180 minus their sum.
  • All angles must be positive and less than 180 degrees individually.
  • An exterior angle is always larger than either of the non-adjacent interior angles.
  • In a valid triangle, no single angle can be 180 degrees or more.
  • If given an exterior angle and one remote interior angle, subtract to find the other.