Triangle Sum Theorem Calculator

The Triangle Sum Theorem

The Triangle Sum Theorem (also called the Triangle Angle Sum Theorem) states that the sum of the three interior angles of any triangle is always 180 degrees. This is one of the most fundamental theorems in Euclidean geometry and applies to all triangles regardless of their type or size.

Interior and Exterior Angles

Interior Angles

The three angles inside the triangle. They always sum to exactly 180 degrees.

A + B + C = 180

Exterior Angles

Each exterior angle is supplementary to its adjacent interior angle (they sum to 180).

Exterior = 180 - Interior

Exterior Angle Theorem

Each exterior angle equals the sum of the two non-adjacent (remote) interior angles.

Ext. A = B + C

Types of Triangles by Angles

Acute Triangle

All three interior angles are less than 90 degrees. For example, a triangle with angles 60, 70, and 50 degrees is acute.

Right Triangle

One interior angle is exactly 90 degrees. The other two angles must sum to 90 degrees. For example, 90, 45, and 45 degrees.

Obtuse Triangle

One interior angle is greater than 90 degrees. The other two must be acute. For example, 120, 35, and 25 degrees.

Equilateral Triangle

All three angles are equal at 60 degrees each (60 + 60 + 60 = 180). All sides are also equal.

Key Properties

The sum of interior angles is always 180 degrees in Euclidean geometry.
The sum of all three exterior angles (one at each vertex) is always 360 degrees.
An exterior angle is always greater than either of the non-adjacent interior angles.
No interior angle of a triangle can be 0 degrees or 180 degrees.
If two angles of one triangle equal two angles of another, the triangles are similar (AA criterion).
The largest angle is always opposite the longest side.