Angle Sum Calculator
The interior angles of any simple polygon with n sides always add up to (n − 2) × 180°.
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How to use this calculator
- Enter the number of sides (n) of your polygon — 3 for a triangle, 4 for a quadrilateral, and so on.
- Press Calculate to see the interior-angle sum, exterior-angle sum, and the per-angle values for a regular polygon.
- Use the interior sum for any polygon; use the per-angle results only when every angle is equal (a regular shape).
How it works
The interior angles of any simple polygon with n sides always add up to (n − 2) × 180°. You can picture this by splitting the polygon into n − 2 triangles from one vertex; each triangle contributes 180°. This holds whether or not the polygon is regular, as long as it does not cross over itself.
The exterior angles of any convex polygon always sum to 360°, no matter how many sides it has. For a regular polygon (all sides and angles equal), each interior angle is the interior sum divided by n, and each exterior angle is 360° ÷ n. The interior and exterior angle at each vertex always add to 180°.
Worked example
Hexagon (n = 6). A six-sided polygon has an interior-angle sum of (6 − 2) × 180° = 720°. If it is regular, each interior angle is 720° ÷ 6 = 120°, and each exterior angle is 360° ÷ 6 = 60°.
Common mistakes
- Using the per-angle result for an irregular polygon — 'each interior angle' only applies when all angles are equal (a regular polygon). The total interior sum, however, is valid for any simple polygon.
- Forgetting that n must be a whole number of at least 3 — there is no polygon with 2 sides or a fractional side count.
- Confusing interior and exterior angles: the exterior angles always total 360°, while the interior total grows by 180° for each extra side.
Frequently asked questions
What is the sum of the interior angles of a polygon?
For a polygon with n sides it is (n − 2) × 180°. A triangle gives 180°, a quadrilateral 360°, a pentagon 540°, and so on — each extra side adds 180°.
Do the exterior angles always add up to 360°?
Yes — for any convex polygon the exterior angles sum to 360° regardless of the number of sides. That is why each exterior angle of a regular polygon is 360° ÷ n.
Does this work for irregular polygons?
The interior-angle sum, (n − 2) × 180°, works for any simple (non-self-intersecting) polygon, regular or not. The 'each angle' outputs assume a regular polygon where all angles are equal.
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