Calculate exterior angles for triangles and regular polygons. Uses the exterior angle theorem for triangles and the 360°/n formula for polygons. Shows angle sums, interior-exterior pairs, and visua...
An exterior angle of a polygon is formed by one side and the extension of an adjacent side. The exterior angle theorem is one of the most elegant results in geometry: for any convex polygon, the sum of all exterior angles (one at each vertex) is always exactly 360°, regardless of the number of sides.
For triangles, the exterior angle theorem has a particularly useful corollary: an exterior angle equals the sum of the two non-adjacent (remote) interior angles. So if a triangle has interior angles of 40°, 60°, and 80°, the exterior angle at the 80° vertex is 40° + 60° = 100°. This is equivalent to 180° − 80° = 100°, since each exterior angle is supplementary to its adjacent interior angle.
For regular polygons, every exterior angle has the same measure: 360°/n, where n is the number of sides. A regular hexagon has exterior angles of 60°, a regular octagon has 45°, and so on. This formula provides a quick way to compute angles without needing to know interior angles first.
This calculator supports two modes. In triangle mode, you enter two remote interior angles and the tool finds the exterior angle, the third interior angle, and all three interior-exterior pairs. In polygon mode, you enter the number of sides and the tool computes each exterior angle, each interior angle, and their relationship. Presets, a comparison table, and visual proportion bars make the concept easy to explore and understand.
The Exterior Angle Calculator — Triangles & Polygons is useful when you need fast and consistent geometry results without reworking the same algebra repeatedly. It helps you move from raw measurements to Exterior Angle, Adjacent Interior Angle, Interior Angle Sum in one pass, with conversions and derived values shown together.
Use it to validate homework steps, check CAD or fabrication dimensions, estimate material requirements, and sanity-check hand calculations before submitting work.
Triangle exterior angle: ext = remote₁ + remote₂ = 180° − adjacent interior Polygon exterior angle (regular): ext = 360° / n Sum of exterior angles (any convex polygon): 360° Interior + Exterior = 180° (supplementary) Interior angle (regular): (n − 2) × 180° / n
Result: Exterior angle = 120°, Adjacent interior = 60°, Sum check = 180°
The two remote interior angles are 50° and 70°. By the exterior angle theorem, the exterior angle = 50° + 70° = 120°. The adjacent interior angle = 180° − 120° = 60°. Sum of all interior angles: 50° + 70° + 60° = 180° ✓.
This calculator combines the core geometry formula with the input mode selected in the interface, then derives companion values shown in the output cards, comparison bars, and reference tables. Use it to cross-check both direct calculations and reverse-solving scenarios where one measurement is unknown.
Exterior Angle Calculator — Triangles & Polygons calculations show up in coursework, drafting, construction layout, packaging, tank sizing, machining, and quality control. Instead of solving each transformation manually, you can test scenarios quickly and verify whether your dimensions remain within tolerance.
Keep units consistent across every input before interpreting area, perimeter, angle, or volume outputs. For best results, measure carefully, round only at the final step, and compare at least one manual calculation with the calculator output when building confidence.
The exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. For example, if the remote angles are 50° and 70°, the exterior angle is 120°.
Imagine walking along the perimeter. At each vertex, you turn by the exterior angle. After a complete loop, you face the same direction — having turned a full 360°.
Divide 360° by the number of sides: exterior angle = 360°/n. For a hexagon (6 sides): 360°/6 = 60°.
They are supplementary: interior + exterior = 180° at each vertex. If the interior angle is 120°, the exterior angle is 60°.
For convex polygons, all exterior angles are between 0° and 180°. Concave (non-convex) polygons can have reflex interior angles, making some exterior angles negative by convention.
It decreases: 360°/3 = 120° for a triangle, 360°/6 = 60° for a hexagon, 360°/36 = 10° for a 36-gon. As n → ∞, the exterior angle → 0° and the polygon approaches a circle.