Interior Angle of Polygon Calculator – Regular and Irregular Polygons
The interior angle of a polygon is the angle formed inside the shape at each vertex, measured between two adjacent sides. For a regular polygon — one where all sides and angles are equal — every interior angle has the same measure. For an irregular polygon, the angles can differ, but their sum is always the same fixed value that depends only on the number of sides.
The Core Formulas
Two foundational formulas govern interior angles of polygons:
- Sum of interior angles:
S = (n − 2) × 180°, where n is the number of sides. - Each interior angle (regular polygon only):
angle = (n − 2) × 180 / n
The sum formula works because any convex polygon with n sides can be divided into n − 2 triangles by drawing non-crossing diagonals from one vertex. Since every triangle contains 180°, the total is always (n − 2) × 180.
Common Polygon Reference
Applying the formulas to the most frequently encountered polygons:
- Triangle (n = 3): sum = 180°; each angle in a regular triangle = 60°
- Quadrilateral (n = 4): sum = 360°; each angle in a square = 90°
- Pentagon (n = 5): sum = 540°; each angle in a regular pentagon = 108°
- Hexagon (n = 6): sum = 720°; each angle in a regular hexagon = 120°
- Octagon (n = 8): sum = 1080°; each angle in a regular octagon = 135°
- Decagon (n = 10): sum = 1440°; each angle in a regular decagon = 144°
Exterior Angles
The exterior angle at a vertex is the angle formed between one side of the polygon and the extension of the adjacent side. For any convex polygon, the exterior angles always sum to exactly 360°. For a regular n-gon:
Each exterior angle = 360 / n
Interior and exterior angles at the same vertex are supplementary — they add to 180°. A regular hexagon has exterior angles of 60°, and a square has exterior angles of 90°.
Finding a Missing Angle in an Irregular Polygon
When working with an irregular polygon where most angles are known but one is missing, the sum formula still applies. If you know the number of sides n and the sum of all known angles K, the missing angle is:
Missing angle = (n − 2) × 180 − K
For example, suppose a pentagon has four angles measuring 95°, 110°, 120°, and 115°. Their sum is 440°. The total interior angle sum of a pentagon is 540°, so the missing angle is 540 − 440 = 100°.
Degrees and Radians
Angles can be expressed in degrees or radians depending on the application. Degrees are standard in geometry education; radians are preferred in advanced mathematics, physics, and programming (where trigonometric functions expect radians by default). The conversion is:
radians = degrees × π / 180
A regular hexagon's interior angle of 120° equals 2π/3 ≈ 2.0944 rad. This calculator lets you toggle between both units instantly.
Number of Diagonals
As a bonus, this calculator also displays the number of diagonals for any polygon. A diagonal connects two non-adjacent vertices. The count is:
Diagonals = n × (n − 3) / 2
A triangle has 0 diagonals; a hexagon has 9; a decagon has 35. This formula grows quadratically — large polygons have enormous numbers of diagonals.
Practical Applications
Architecture and Design
Architects and interior designers frequently work with polygonal rooms, tiles, and facades. Knowing that a regular octagon's interior angles are 135° ensures corner joints fit flush. Hexagonal tiles, used in mosaic patterns and honeycomb designs, tile perfectly because their 120° angles sum to 360° at every meeting point.
Engineering and CAD
Structural engineers designing polygonal frames need interior angles to compute stress distributions. CAD software often requires explicit angle inputs for polygon primitives, and this calculator provides instant values for any n without manual arithmetic.
Education
Students learning geometry can verify results for homework problems, explore how angles grow as n increases (approaching 180° as the polygon approaches a circle), and use the step-by-step breakdown to understand the derivation.
Key Observations
- As n increases, each interior angle of a regular polygon approaches 180° — the polygon becomes increasingly circular.
- The sum of interior angles grows by 180° with every additional side.
- All regular polygons with the same n are similar — they have identical angles regardless of size.
- A polygon with all interior angles less than 180° is convex; one with at least one angle exceeding 180° is concave (the sum formula still applies for concave simple polygons, but individual angles can be reflex).