Interior and Exterior Triangle Angles Calculator

Understanding Interior Angles

Interior angles are the three angles inside a triangle at its vertices. The triangle angle sum theorem—one of geometry's most fundamental principles—states that these three angles always total 180° (π radians in radian measure).

If you know two interior angles, finding the third is straightforward: subtract both known angles from 180°. For example, if a triangle has interior angles of 60° and 70°, the third must be 180° − 60° − 70° = 50°.

This theorem applies universally: acute triangles, right triangles, obtuse triangles—the sum never changes. It's a cornerstone of Euclidean geometry and the foundation for calculating exterior angles.

Exterior Angles Explained

An exterior angle forms when you extend any side of a triangle beyond its vertex. At each of the three vertices, you can form two exterior angles (one on either side), giving six exterior angles total. Importantly, the two exterior angles at any single vertex are equal because they're vertically opposite.

The exterior angle theorem reveals a powerful relationship: any exterior angle equals the sum of the two non-adjacent (opposite) interior angles. If interior angles are A, B, and C, then the exterior angle adjacent to C equals A + B.

Since exterior angles at the same vertex are equal, this effectively gives you three distinct exterior angle values. The six exterior angles always sum to 360° because three pairs of 180° angles (straight lines) equals three times 360° ÷ 3 = 120° each on average, though individually they vary by triangle shape.

Key Formulas for Triangle Angles

Use these relationships to calculate unknown angles. The triangle angle sum theorem finds missing interior angles, while the exterior angle theorem connects exterior angles to opposite interior angles.

• A, B, C — Interior angles of the triangle
• D, E, F — Exterior angles adjacent to interior angles A, B, C respectively

Common Angles and Practical Tips

When working with triangle angles, keep these practical insights in mind to avoid calculation errors and understand geometric relationships.

1. Right triangles simplify calculations — In a right triangle, one angle is always 90°. If you know the second angle, subtract both from 180° to get the third. For instance, with a 90° and 35° angle, the third is 55°. This makes right triangles easier to work with in construction and engineering.
2. Exterior angles are always larger than non-adjacent interior angles — An exterior angle must exceed each of the two opposite interior angles individually because it equals their sum. This property helps verify your calculations—if an exterior angle is smaller than one of the opposite interior angles, you've made an error.
3. Radians and degrees require unit consistency — 180° equals π radians. If your triangle uses radian inputs, ensure all angle values stay in radians. Many calculation errors stem from mixing units. Convert beforehand: multiply degrees by π/180 to get radians, or multiply radians by 180/π for degrees.
4. Exterior angle pairs at one vertex are identical — Don't double-count exterior angles. At each vertex, the two exterior angles formed on opposite sides of the extended line are equal. You really have three distinct exterior angles, not six different values, even though six exterior angles exist geometrically.

Using the Calculator

Input any two of the three interior angles in your preferred units (degrees or radians). The calculator immediately computes the missing interior angle using the triangle angle sum theorem, then derives all six exterior angles via the exterior angle theorem.

The tool displays interior angles A, B, and C, plus the three distinct exterior angles D, E, and F. If you need to verify your own geometry work or explore how changing one angle affects the others, experiment with different angle pairs—the relationships remain mathematically consistent.