Separate the angle sum from one regular angle
The interior angle sum depends only on the side count of a simple polygon; individual angles need not be equal. This polygon angle calculator divides that sum by n only for the explicitly regular result, where every side and interior angle is equal. Keeping the outputs separate prevents an irregular or concave shape from being described as though every corner had the regular value.
The polygon angle calculator accepts a whole-number side count, a named preset, or a test of whether a stated regular interior, exterior, or central angle corresponds closely to integer n. Up to 24 sides, the SVG creates focusable vertices. Larger values use a simplified circle so the page never draws thousands of labels.
How to read the polygon results
- Enter n as an integer from 3 through 10,000. Fractions, values below three, and oversized values are rejected instead of silently rounded.
- Read Interior angle sum in the polygon angle calculator for any simple n-sided polygon, regular or irregular. It totals all interior corners.
- Use One regular interior angle only when the polygon is regular. Exterior and central outputs also describe the equal-angle regular construction.
- Read diagonals as connections between nonadjacent vertices and triangles from one vertex as the triangulation behind the sum formula.
- Show radians when needed, then copy, share, or print the formula substitution and result.
Polygon formulas and their scope
- Interior angle sum = (n − 2) × 180°. Diagonals from one vertex divide a simple n-gon into n − 2 triangles.
- One regular interior angle = ((n − 2) × 180°) ÷ n. This division is not valid for a general irregular polygon.
- One regular exterior angle = 360° ÷ n because consistent exterior turns complete a full rotation.
- The central angle between adjacent regular vertices is also 360° ÷ n, though it is drawn at the center rather than an extended side.
- Diagonals = n(n − 3) ÷ 2. Each vertex reaches n − 3 nonadjacent vertices, and division by two removes double counting.
Worked polygon examples
- Triangle, n = 3: the sum is 180°. A regular triangle has 60° interior angles, 120° exterior and central angles, and no diagonals.
- Square, n = 4: the sum is 360°, each regular interior angle is 90°, exterior and central angles are 90°, and there are two diagonals.
- Regular pentagon, n = 5: the sum is 540°, one interior angle is 108°, exterior and central angles are 72°, and there are five diagonals.
- Regular octagon, n = 8: the sum is 1,080°, one interior is 135°, exterior and central are 45°, and diagonals total 20.
- A 100-sided polygon has a 17,640° sum. If regular, each interior is 176.4° and exterior or central is 3.6°; its drawing is simplified.
Interior, exterior, and central locations
An interior angle lies inside between adjacent sides. The exterior angle here is the turn between a side and the forward extension of its neighbor. A central angle lies between radii to adjacent vertices. Exterior and central values both equal 360°/n in a regular polygon but occupy different regions.
The inverse helper suggests n only when the side count is within a strict tolerance of a legal integer. It never forces 5.93 sides into a hexagon.
A limited, accurate name list
The interface maps familiar names such as triangle, square, pentagon, hexagon, heptagon, octagon, nonagon, decagon, and dodecagon. Other results use the plain description n-sided polygon rather than inventing or misspelling a rare name.
Simple, regular, irregular, and concave polygons
The sum formula applies to simple polygons whose edges do not cross. Self-intersecting star polygons need winding-specific treatment. A concave simple polygon keeps the same sum but its individual angles are unequal and at least one exceeds 180°. The SVG is an ideal regular illustration, not a scale drawing, mesh validator, area calculator, or proof that a real shape is regular.
Calculation is browser-side. A shared URL stores only n and whether radians are visible; no drawing or document is uploaded.