math

Triangle Sum Theorem Calculator

Every triangle's interior angles always sum to exactly 180°. Whether you're solving a geometry problem, verifying a construction, or working through a proof, this tool quickly determines any missing angle when you know the other two. Enter your known angles in degrees or radians and get the answer in seconds—no manual arithmetic required.

Last updated: May 15, 2026

Creators Anna Szczepanek, PhD Mathematics

Reviewers Mateusz Mucha and Jasmine J. Mah

3,158 people find this calculator helpful

Understanding the Triangle Angle Sum

The triangle angle sum theorem is one of geometry's most fundamental rules: the three interior angles of any triangle always add up to 180 degrees (or π radians). This holds true regardless of the triangle's shape, size, or type—whether it's equilateral, isosceles, right-angled, or scalene.

This principle forms the foundation for countless geometry problems. If you know any two angles, the third becomes straightforward to find. Simply subtract both known angles from 180°, and you have your answer.

The theorem applies universally in Euclidean geometry, making it essential knowledge for students, engineers, architects, and anyone working with shapes and spatial relationships.

Calculating the Missing Angle

When two angles of a triangle are known, rearrange the basic theorem to isolate the unknown angle. The formula below shows how to find any missing angle:

α = 180° − β − γ

α = π − β − γ (in radians)

α (alpha) — The unknown or target angle in degrees or radians
β (beta) — The first known interior angle
γ (gamma) — The second known interior angle

How the Calculator Works

Input any two interior angles of your triangle—the calculator accepts values in degrees, radians, or gradians depending on your preference. Once submitted, it instantly computes the third angle using the angle sum formula.

The tool handles the conversion and arithmetic automatically, eliminating human error. You can verify solutions for homework, check your manual calculations, or quickly solve multi-step geometry problems where finding missing angles is one component.

The calculator also works in reverse: if you know all three angles and need to verify they form a valid triangle, simply check that they sum to 180°.

Common Pitfalls and Considerations

Keep these practical points in mind when working with triangle angles.

Verify the angle sum before drawing conclusions — Always confirm that your two known angles add up to less than 180°. If they sum to 180° or more, no valid triangle exists with those angles. This is a quick sanity check that catches data entry errors immediately.
Watch your angle units — Mixing degrees and radians is a frequent source of mistakes. The calculator handles conversions, but manually entering mixed units will produce nonsense results. Pick one system and stick with it throughout your calculation.
Remember the constraints for specific triangle types — While any three angles summing to 180° form a valid triangle mathematically, real-world applications may require additional constraints. Right triangles must include a 90° angle; equilateral triangles have three 60° angles. Verify your result matches the triangle type you need.

Practical Example

Suppose you're working with a triangle where one angle measures 40° and another is 75°. To find the third angle:

Step 1: Add the two known angles: 40° + 75° = 115°

Step 2: Subtract from 180°: 180° − 115° = 65°

The third angle is 65°. You can verify: 40° + 75° + 65° = 180° ✓

In construction, surveying, or navigation, this calculation often appears when measuring angles at different points or verifying structural geometry.

Frequently Asked Questions

What is the triangle angle sum theorem?

The triangle angle sum theorem states that the sum of all three interior angles in any triangle equals exactly 180 degrees (π radians). This universal property holds for every triangle, regardless of its size, orientation, or classification. The theorem is one of the most important principles in Euclidean geometry and forms the basis for countless geometric proofs and practical applications in engineering and construction.

Can a triangle have two angles that are both obtuse?

No, a triangle cannot have two obtuse angles. An obtuse angle is any angle greater than 90°. If two angles each exceeded 90°, their sum alone would surpass 180°, leaving no room for a third positive angle. A triangle can have at most one obtuse angle; the other two must be acute (less than 90°).

What do the angles sum to if I'm working with radians instead of degrees?

When using radians, the three interior angles of any triangle sum to exactly π radians instead of 180°. This is the same 180-degree equivalence expressed in radian measure. To solve for a missing angle in radians, use the formula: α = π − β − γ. If you have angles in degrees, convert them to radians first by multiplying by π/180.

How do I know if three angles can form a real triangle?

For three angles to form a valid triangle, they must sum to exactly 180° (or π radians in radian measure), and each angle must be positive and less than 180°. If the sum is less than or greater than 180°, no triangle exists. If any angle equals zero or reaches 180°, you don't have a proper triangle—it degenerates into a line.

Why is the triangle angle sum always 180°?

This fundamental property arises from the parallel postulate in Euclidean geometry. If you draw a line parallel to one side of a triangle through the opposite vertex, the angles formed create a straight line (180°). By comparing alternate interior angles, you can show the three angles of the original triangle must sum to 180°. This is one of the foundational truths of flat-plane geometry.

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