Polygon Angle Calculator

How to use the polygon angle calculator

What this calculator does

This calculator converts the number of sides in a regular convex polygon into five connected angle results: Interior angle α, Exterior angle β, Sum of interior angles, Sum of exterior angles, and Interior + exterior. It is useful for checking geometry exercises, planning regular shapes in design or fabrication, and confirming the turning angle used to draw an equal-sided polygon. The calculator assumes every side and every corresponding angle is equal. It does not determine the individual angles of an irregular polygon, a concave polygon, or a self-intersecting star polygon.

When to use it

• Check the interior and exterior angle of a named regular polygon such as a pentagon, hexagon, or dodecagon.
• Find the turn required after drawing each side in a regular-polygon construction.
• Verify the total interior-angle sum before solving for missing angles in a polygon problem.
• Switch a classroom result between degrees and radians without changing the polygon itself.

How to calculate

The calculator opens with a complete six-sided example, so the results and the example Excel workbook are available immediately.

1. Replace Number of sides with the whole-number side count of the regular polygon you want to study.
2. Choose Angle unit: degrees for most geometry work or radians for scientific and technical calculations.
3. Read the live result cards and the calculation-detail table. The summary pills identify the polygon, confirm that α + β is one straight angle, and show the constant exterior-angle total.
4. Select Download Excel to create a current-state workbook containing the inputs, outputs, formulas, units, and explanatory notes.
5. Select Reset to clear the demonstration input. Reset does not restore six sides; it leaves the required field empty and may disable Download Excel until a complete valid side count is entered again.

Input guide

Number of sides is required and accepts an ungrouped whole number from 3 through 1,000,000, such as 8. A polygon needs at least three sides. Increasing the side count makes each interior angle approach a straight angle while each exterior angle becomes smaller. Entering a decimal, comma-grouped number, scientific notation, negative value, or text produces a visible validation message rather than a silent reinterpretation.

Angle unit is required and offers Degrees (°) or Radians (rad). Changing it converts every displayed and exported angle while preserving the same geometry. For example, 120° is 2.0943951 rad. A common mistake is to compare a degree result directly with a radian result as though the numbers used the same scale. NIST identifies the radian as the coherent SI unit for plane angle and gives the exact degree conversion in its SI conversion table for plane angle.

Output guide

Interior angle α is the angle inside one vertex of the regular polygon. It is an exact consequence of the side count, displayed in the selected unit. The minimum occurs for a triangle; as the side count grows, α approaches but never reaches 180°. Exterior angle β is the adjacent turning angle and also equals the central angle between neighboring vertices. It is always positive and decreases as the number of sides increases.

Sum of interior angles totals all interior vertex angles and grows by 180° for each added side. Sum of exterior angles is always one full turn – 360° or 2π rad – for a convex polygon when one consistent exterior angle is taken at every vertex. Interior + exterior confirms the supplementary identity α + β = 180° or π rad. The Polygon pill gives a familiar name for common side counts and otherwise uses the standard “n-gon” form. The detail table repeats each output with its applied formula; its rows and the Excel workbook are generated from the same canonical values.

Worked example

For the startup example, n = 6 and the unit is degrees. The interior-angle formula gives α = (6 – 2) × 180° ÷ 6 = 720° ÷ 6 = 120°. The exterior-angle formula gives β = 360° ÷ 6 = 60°. Multiplying the interior angle by six gives an interior sum of 720°; multiplying the exterior angle by six gives 360°. Finally, 120° + 60° = 180°, confirming that the adjacent interior and exterior angles form a straight angle. OpenStax presents the same regular-polygon formulas and worked geometry method in its polygons, perimeter, and circumference chapter.