Exterior angles are one of those “secret weapons” topics in geometry: once you understand them, a lot of polygon problems become almost unfairly easy. Whether you’re working with triangles, pentagons, stop‑sign octagons, or a 30‑sided polygon that sounds like a Pokémon, the exterior angle idea gives you a clean shortcut: **the sum of one exterior angle at each vertex of any polygon is always 360°**. This page gives you a fast Exterior Angles Calculator (regular polygons + quick conversions from interior angles), plus a full explanation so you can understand the math, not just copy the answer. --- ## What is an exterior angle? At a vertex (corner) of a polygon, you can measure angles in two common ways: - **Interior angle**: the angle *inside* the polygon. - **Exterior angle**: the angle you get when you extend one side of the polygon and measure the “turn” you make to follow the next side. Think of walking around the polygon’s boundary. Every time you hit a corner, you rotate a little to stay on the perimeter. That rotation is the **exterior angle**. If you go all the way around and return to your starting direction, you’ve made a full turn—**360°**. That’s the intuition behind the “sum is 360°” rule. There are two common exterior angle conventions: 1) **The turning angle** (usually the “outside” turn as you trace the polygon). 2) **The supplementary exterior angle** (sometimes teachers call the outside angle that forms a linear pair with the interior angle). In this calculator, we use the standard “turning” exterior angle that pairs with the interior angle of a convex polygon: > **Interior angle + Exterior angle = 180°** (for convex polygons, using the adjacent exterior angle). So if you know one, you can get the other instantly. --- ## The two “always works” facts ### 1) Sum of exterior angles = 360° If you take **one exterior angle at each vertex** (always turning the same direction), then: **Sum of exterior angles = 360°** This is true for: - Triangles, quadrilaterals, pentagons, etc. - Regular and irregular polygons - Any convex polygon (and still works for many non‑convex polygons if you’re careful with signed turns) This is one of the cleanest constants in geometry: no matter how many sides your polygon has, the total turning you do is a full circle. ### 2) For a regular polygon, each exterior angle is equal A **regular polygon** has all equal sides and all equal angles. That means every turn is the same, so: **Each exterior angle = 360° / n** where **n** is the number of sides. Example: regular hexagon (n = 6) Each exterior angle = 360° / 6 = 60°. Then the interior angle is: Interior = 180° − Exterior = 180° − 60° = 120°. --- ## What this calculator can do This Exterior Angles Calculator supports the most common needs: ### A) Regular polygon exterior angle (from number of sides) If you enter **n**, it returns: - Each exterior angle = 360° / n - Sum of exterior angles = 360° - Each interior angle = 180° − (360° / n) - Sum of interior angles = (n − 2) × 180° ### B) Exterior angle from interior angle (regular polygon or convex corner) If you enter an **interior angle**, it returns: - Exterior angle = 180° − interior angle If you also provide **n** and it’s a regular polygon, the calculator will check consistency (interior implied by n matches your entered interior) and tell you if your inputs disagree. ### C) Number of sides from exterior angle (regular polygon) If you enter an exterior angle for a regular polygon: - n = 360° / exterior If 360 is not divisible cleanly, the calculator will show a decimal and explain that a perfect regular polygon typically expects an integer n. (In real geometry problems, they usually choose numbers that divide nicely.) --- ## Formula breakdown (with meaning) ### 1) Sum of interior angles For any polygon with **n** sides: **Sum of interior angles = (n − 2) × 180°** Why? A polygon can be divided into **(n − 2)** triangles by drawing diagonals from one vertex. Each triangle has 180°, so multiply. ### 2) Each interior angle (regular polygon only) If the polygon is regular: **Each interior angle = ((n − 2) × 180°) / n** ### 3) Each exterior angle (regular polygon only) Still regular: **Each exterior angle = 360° / n** ### 4) Relationship between interior and exterior At each vertex (convex, adjacent exterior): **Interior + Exterior = 180°** So: - Exterior = 180° − Interior - Interior = 180° − Exterior These four formulas cover almost every “find the angle / find n” problem you’ll see in school geometry. --- ## Step-by-step examples ### Example 1: Find the exterior angle of a regular octagon An octagon has n = 8. Exterior = 360° / 8 Exterior = 45° So each turn is 45°. Interior = 180° − 45° = 135°. ### Example 2: Find the number of sides if each exterior angle is 24° For a regular polygon: n = 360° / 24° n = 15 So it’s a regular 15‑gon. ### Example 3: Interior angle is 156°. What is the exterior angle? Exterior = 180° − 156° Exterior = 24° This matches Example 2 (regular 15‑gon), because if the exterior is 24°, n = 15. ### Example 4: Quick check—does a regular polygon exist with exterior angle 22°? n = 360° / 22° ≈ 16.3636… Since n is not an integer, a perfect regular polygon with exactly 22° exterior angles is not a standard regular polygon. In many textbook problems, they would not choose 22° unless the question expects a non‑integer n (rare). So this is a great “sanity check” tool. ### Example 5: Irregular polygon sum of exterior angles Even if the polygon is irregular (different sides/angles), as long as you take one exterior angle at each vertex and turn consistently: Sum = 360°. So if you know four exterior angles of a pentagon are 70°, 90°, 40°, and 60°, the fifth is: Fifth = 360° − (70 + 90 + 40 + 60) Fifth = 360° − 260 Fifth = 100°. This calculator focuses on regular polygon computations and interior↔exterior conversions, but that 360° idea is the backbone for “missing exterior angle” problems too. --- ## How it works (what the code is doing) 1) **Validate inputs** - n must be ≥ 3 - angles should be between 0° and 180° (for convex use) 2) **Pick the strongest route** - If n is present → compute regular polygon values from n - Else if exterior angle is present → compute n = 360/exterior (regular assumption) - Else if interior angle is present → compute exterior = 180 − interior 3) **Compute and display** The results include a clean sentence (good for screenshots) plus “details” lines that explain where numbers come from. 4) **Share and save** Just like the Soulmate template, you can copy/share the results. Saved results are stored locally in your browser (no server). --- ## Common mistakes (and how to avoid them) ### Confusing “sum of exterior angles” with “sum of interior angles” - Exterior sum is always **360°** (one at each vertex, same direction). - Interior sum is **(n − 2) × 180°**, which grows as n grows. ### Using the wrong exterior angle definition Sometimes books define the exterior angle as the angle outside the polygon that forms a linear pair with the interior angle (adjacent). That’s the one we use here, so: Interior + Exterior = 180°. If your class is using a different convention (like reflex exterior angles), you’ll need a different relationship. For most school geometry with convex polygons, our definition is the standard. ### Forgetting regular vs irregular - **Each exterior angle = 360/n** only for **regular** polygons. - But **sum of exterior angles = 360°** works for any polygon (one per vertex). ### Not checking divisibility If an exterior angle doesn’t divide 360 nicely, it usually means: - It’s not a regular polygon problem, or - The angle given isn’t the “turning” exterior angle, or - The problem expects a non‑integer n (rare) --- ## FAQs ### Is the sum of exterior angles always 360°? Yes—if you take one exterior angle at each vertex and “walk” around the polygon in one consistent direction, you make a full 360° turn. This is true for all convex polygons and still works for many non‑convex shapes when you use signed turns. ### What’s the exterior angle of a triangle? For a regular (equilateral) triangle, each exterior angle is 360/3 = 120°. For a general triangle, the exterior angle at a vertex equals the sum of the two remote interior angles (another classic theorem), but the turning‑angle sum around the triangle is still 360°. ### How do I find the interior angle from n? If it’s a regular polygon: Interior = 180° − 360°/n. Or equivalently: Interior = ((n − 2) × 180°) / n. ### How do I find n from the interior angle? Regular polygon: Exterior = 180° − interior n = 360° / exterior. ### Why do regular polygons have equal exterior angles? Because each side length and corner is identical, the “turn” at each vertex is identical. If the total turning is 360°, dividing it equally across n corners gives 360/n. ### Can an exterior angle be greater than 180°? For convex polygons using the adjacent exterior angle, it will be between 0° and 180°. For non‑convex polygons, you can have reflex turns and you’d treat those as signed angles. This calculator is optimized for the convex/regular cases you see most often. ### What is a real-world use for exterior angles? Exterior angles show up in: - Architecture and tiling patterns (regular polygons) - Computer graphics and robotics (turning angles along a path) - Navigation/path planning (total turning) - Design of signs, tables, and decorative patterns that use regular n‑gons --- If you want a “quick win”: memorize **360°/n** for regular polygons and **180° − interior** for conversions. With those two, you can solve most exterior angle problems in seconds.