Angle Sum Property of a Triangle Calculator
Enter two angles of any triangle to find the third, or verify that three angles form a valid triangle.
📐 What is the Angle Sum Property of a Triangle?
The angle sum property of a triangle is one of the most fundamental results in Euclidean geometry: the three interior angles of any triangle always add up to exactly 180 degrees. Whether the triangle is equilateral, isosceles, scalene, right-angled, acute, or obtuse, the rule holds without exception. If the three angles are labelled A, B, and C, then A + B + C = 180°. This simple relationship lets you find any one angle as long as you know the other two.
The most intuitive proof uses parallel lines. Draw a line through the apex of the triangle parallel to its base. Two pairs of alternate interior angles are formed. By the parallel-lines theorem, each alternate interior angle equals the corresponding base angle of the triangle. The three angles clustered at the apex (the two alternate interior angles and the apex angle itself) form a straight line, which measures 180°. Therefore the three interior angles of the triangle must also sum to 180°.
This property has immediate practical uses. Roof designers use it to calculate rafter angles from the known pitch angle and the horizontal base. Land surveyors close triangulation loops by verifying that measured angles sum to 180°; any discrepancy indicates measurement error. Structural engineers check truss geometry using angle sums to ensure frames are truly triangular. Even video game developers apply the angle sum property when constructing meshes and testing whether polygon vertices form proper triangles.
Common errors arise when students confuse interior and exterior angles, or forget that the rule applies only in flat (Euclidean) geometry. On a sphere, triangle angles sum to more than 180°, which matters in large-scale navigation and geodesy. On a hyperbolic surface they sum to less than 180°. For everyday classroom geometry and most real-world applications, however, the Euclidean 180° rule is exact and universally applicable.
This calculator handles two tasks: finding the unknown third angle from two known angles, and verifying that a set of three angles is geometrically consistent. In Find Angle mode, enter any two interior angles and the calculator instantly computes the third using C = 180° − A − B. In Check Triangle mode, enter all three angles and the tool confirms whether they sum to 180° and classifies the triangle. Both modes also identify the triangle type — acute, right, or obtuse — based on the largest angle present.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Equilateral Triangle
All three angles are equal. Using A = B = C and A + B + C = 180°: each angle = 180° ÷ 3 = 60°. Enter A = 60°, B = 60° → C = 180° − 60° − 60° = 60°. Triangle type: acute (all angles < 90°).
Example 2 — Right Triangle
A right triangle has one 90° angle. Suppose A = 90°, B = 35°. Then C = 180° − 90° − 35° = 55°. The sum is 90° + 35° + 55° = 180° ✓. Triangle type: right (one 90° angle). Angles B and C are complementary (35° + 55° = 90°).
Example 3 — Obtuse Triangle
Suppose A = 120° and B = 30°. Then C = 180° − 120° − 30° = 30°. Sum: 120° + 30° + 30° = 180° ✓. Triangle type: obtuse (angle A = 120° > 90°). This is also an isosceles obtuse triangle since B = C.
Example 4 — Check Mode: Valid Angles
Enter A = 47.5°, B = 82.3°, C = 50.2°. Sum = 47.5 + 82.3 + 50.2 = 180.0° — valid triangle ✓. Largest angle is 82.3° < 90°, so the triangle is acute.