Calculate all interior and exterior angles of a triangle from 3 sides (SSS) or 2 known angles. Verifies angle sum = 180° and exterior sum = 360°. Includes remote interior angle theorem, presets, an...
<p>The <strong>Interior & Exterior Triangle Angles Calculator</strong> computes all three interior angles and their corresponding exterior angles from either three side lengths (using the law of cosines) or two known angles (since the three must sum to 180°). This is one of the most fundamental operations in plane geometry.</p> <p>Every triangle has an interior angle sum of exactly 180° and an exterior angle sum of 360°. The exterior angle at each vertex equals 180° minus the interior angle and also equals the sum of the two non-adjacent (remote) interior angles — this is the <em>Exterior Angle Theorem</em>. This calculator demonstrates all three properties with clear output cards and verifications.</p> <p>In the SSS mode, the law of cosines is applied: cos A = (b² + c² − a²) / (2bc), where a is the side opposite angle A. The calculator first checks the triangle inequality to ensure a valid triangle exists, then computes all angles in degrees.</p> <p>Whether you are studying geometry proofs, surveying land, or designing structures, this tool provides instant angle calculations complete with classification (acute, right, obtuse), a visual angle comparison bar chart, presets for common triangles (equilateral, 30-60-90, isosceles), and a reference table for standard triangle types.</p>
This calculator is most useful when you need more than a single missing angle. In one step it gives you all three interior angles, all three exterior angles, the interior and exterior sum checks, and a direct verification of the remote interior angle theorem. That makes it practical for geometry proofs, classroom exercises, drafting work, and any triangle-layout problem where you want the full angle picture instead of one isolated answer.
It also reduces a common source of mistakes: mixing up opposite sides, supplementary angles, and remote interior relationships. Whether you start from three sides or two known angles, the tool keeps those relationships synchronized so you can verify diagrams, compare triangle types, and catch impossible measurements quickly.
Law of Cosines: cos A = (b² + c² − a²) / (2bc) Interior angle sum: A + B + C = 180° Exterior angle: ext(A) = 180° − A Exterior angle sum: 360° Exterior Angle Theorem: ext(A) = B + C
Result: A ≈ 38.21°, B ≈ 57.12°, C ≈ 84.67°; exterior angles ≈ 141.79°, 122.88°, and 95.33°.
Using SSS mode with side a = 5, side b = 7, and side c = 8, the calculator applies the law of cosines to each angle. The interior angles add to 180°, and each exterior angle equals 180° minus its matching interior angle. The three exterior angles add to 360°, and ext(A) also matches B + C, confirming the exterior angle theorem.
Each exterior angle in a triangle is supplementary to its interior partner, so the pair always totals 180°. That simple fact lets you move back and forth between the inside and outside view of the same triangle. In proofs and diagrams, this is often the fastest way to check whether a labeled figure is consistent. If one interior angle is 72°, the exterior angle at that same vertex must be 108°.
The calculator makes those paired relationships explicit. Instead of solving an interior angle and then manually converting it, you immediately see both values side by side. That is useful in classroom settings where students are learning angle relationships, but it is also helpful in layout work where angles may be marked on an extension line rather than inside the triangle itself.
When all three side lengths are known, the angle values are not obvious from inspection unless the triangle is a familiar special case. The SSS mode uses the law of cosines to recover each interior angle from the opposite side. This is especially useful for scalene triangles, where no symmetry shortcuts apply.
Once the interior angles are found, the exterior angles follow immediately. The output then checks both global rules at the same time: the interior angles must total 180°, and the exterior angles must total 360°. That combination helps catch bad measurements and invalid side combinations before they make it into homework, CAD drawings, or field notes.
One of the most important triangle facts is that an exterior angle equals the sum of the two remote interior angles. This appears constantly in geometry proofs, standardized test questions, and reasoning problems involving extended sides. Students often memorize the rule but still benefit from seeing it numerically confirmed.
The theorem is also practical outside the classroom. If two interior angles of a triangular frame are known, the turning angle at the third corner can be verified immediately from their sum. The comparison table in this calculator is designed for exactly that kind of check, so you can confirm both the local supplementary relationship and the remote-angle relationship in one place.
The exterior angle at a vertex is formed by one side of the triangle and the extension of the adjacent side. It equals 180° minus the interior angle at that vertex.
Also called the Exterior Angle Theorem, it states that each exterior angle of a triangle equals the sum of the two non-adjacent interior angles. Use this as a practical reminder before finalizing the result.
Yes. This is a fundamental theorem of Euclidean geometry: the angle sum of every triangle is exactly 180°.
Each exterior angle is 180° − interior angle. Summing all three: 3 × 180° − 180° (interior sum) = 360°.
Use the law of cosines: cos A = (b² + c² − a²) / (2bc), then A = arccos(result). Repeat for each angle.
The calculator checks the triangle inequality. If any side is greater than or equal to the sum of the other two, it reports an error — no valid triangle exists.