Interior Angles Calculator — Regular & Custom Polygons

About the Interior Angles Calculator — Regular & Custom Polygons

The interior angles of a polygon are the angles formed inside the shape at each vertex. The sum of all interior angles of an n-sided polygon is given by the elegant formula (n − 2) × 180°. For a triangle (n = 3), the sum is 180°; for a quadrilateral, 360°; for a pentagon, 540°; and so on, increasing by 180° with each additional side.

For regular polygons — where all sides and angles are equal — each interior angle is simply the total sum divided by the number of sides: (n − 2) × 180° / n. A regular hexagon has interior angles of 120°, a regular octagon has 135°, and a regular dodecagon (12 sides) has 150°.

This calculator works in two modes. In regular polygon mode, you enter the number of sides and see each interior angle, the angle sum, the corresponding exterior angle, and how many triangles the polygon can be divided into. In custom polygon mode, you can enter individual angle values for an irregular polygon, and the calculator verifies whether they sum correctly, flags any errors, and computes the missing angle if one is left blank.

The page includes presets for common polygons from triangle through icosagon (20 sides), a reference table comparing angle properties across polygon types, and visual bars showing the interior/exterior angle ratio at each polygon size. It is an essential resource for geometry students, architects designing tiling patterns, and engineers working with multi-sided shapes.

When This Page Helps

The Interior Angles Calculator — Regular & Custom 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 Polygon, Each Interior Angle, Sum of Interior Angles in one pass, with conversions and derived values shown together.

How to Use the Inputs

1. Choose the mode: Regular Polygon or Custom Polygon.
2. For regular polygon mode, enter the number of sides (3 to 100) or click a preset.
3. For custom polygon mode, enter the number of sides, then fill in individual angle values.
4. Leave one angle blank in custom mode and the calculator will find it for you.
5. View the interior angle, angle sum, exterior angle, and number of triangles formed.
6. Use the comparison table to see how angles change from 3 to 20 sides.
7. Check the visual bars showing interior vs. exterior angle proportions.

Example Calculation

Tips & Best Practices

• The formula (n − 2) × 180° works because any polygon can be split into (n − 2) non-overlapping triangles.
• As the number of sides increases, interior angles approach 180° and the polygon approaches a circle.
• Only certain regular polygons tile the plane: equilateral triangle (60°), square (90°), and regular hexagon (120°).
• In custom mode, if your angles don't sum to (n − 2) × 180°, the shape is geometrically impossible.
• The exterior angle of a regular polygon is always 360°/n — a quick alternative to computing the interior angle first.

How This Interior Angles Calculator — Regular & Custom Polygons Works

Where It Helps In Practice

Interior Angles Calculator — Regular & Custom 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.

Accuracy And Setup Tips

Frequently Asked Questions

• What is the formula for the sum of interior angles?

  The sum of interior angles of an n-sided polygon is (n − 2) × 180°. For example, a hexagon: (6 − 2) × 180° = 720°.

• How do I find each interior angle of a regular polygon?

  Divide the total sum by the number of sides: (n − 2) × 180° / n. For a regular pentagon: 540° / 5 = 108°.

• What is the difference between regular and irregular polygons?

  A regular polygon has all sides equal and all angles equal. An irregular polygon has sides and/or angles of different measures. The angle sum formula works for both.

• Why does the sum increase by 180° for each additional side?

  Each additional side adds one more triangle when the polygon is triangulated. Since each triangle contributes 180°, the sum increases by 180° per side.

• Which regular polygons can tile a floor?

  Only three: equilateral triangles (60° × 6 = 360°), squares (90° × 4 = 360°), and regular hexagons (120° × 3 = 360°). The interior angle must divide evenly into 360°.

• What happens as the number of sides approaches infinity?

  Each interior angle approaches 180°, each exterior angle approaches 0°, and the polygon becomes indistinguishable from a circle.