Regular Polygon Calculator

Regular Polygons

A regular polygon has all sides equal and all angles equal.

Formulas

Polygon Properties

Key Relationships

• Sum of Interior Angles: (n-2) × 180°
• Each Interior Angle: (n-2) × 180° / n
• Exterior Angle: 360° / n
• Number of Diagonals: n(n-3)/2

How to Use This Polygon Calculator

1. Set the number of sides (n): pick from triangle (3) up to dodecagon (12) in the dropdown.
2. Enter what you know: choose side length, circumradius, or apothem from the "Calculate from" menu and type its value.
3. Click "Calculate": the tool solves the full geometry — area, perimeter, apothem, circumradius, interior and exterior angles, angle sum, diagonals, and the polygon's name.

Worked Example: Regular Hexagon

A regular hexagon with side length 5:

• Perimeter: P = 6 × 5 = 30
• Area: A = 6 × 5² / (4 × tan(180°/6)) = 150 / (4 × 0.5774) ≈ 64.95 square units
• Interior angle: (6 − 2) × 180° / 6 = 120°
• Exterior angle: 360° / 6 = 60°
• Diagonals: 6 × (6 − 3) / 2 = 9
• Apothem: 5 / (2 × tan(30°)) ≈ 4.33
• Circumradius: 5 / (2 × sin(30°)) = 5 / 1 = 5

Notice the hexagon shortcut: its side length equals its circumradius. Six equilateral triangles fit together to make the hexagon, so R = s holds only for n = 6.

Solving from Different Starting Points

You do not need the side length to start. One compatible measurement is enough:

• From perimeter: divide by n to get the side. A hexagon with P = 30 has s = 30 / 6 = 5.
• From area: rearrange the area formula to recover the side, then compute everything else.
• From apothem or circumradius: use the right triangle formed by the center, a vertex, and a side midpoint — trigonometry converts either radius into the side length.

Angles and Diagonals in More Detail

Each additional side adds 180° to the interior angle sum, so the pattern is predictable:

• Interior angle sum: (n − 2) × 180°. Triangle: 180°, quadrilateral: 360°, hexagon: 720°.
• Exterior angle: 360° / n. These are the turn angles — walking around any polygon turns you a total of 360°.
• Diagonals: n(n − 3)/2. Each of the n vertices connects to n − 3 others (excluding itself and its two neighbors), and dividing by 2 corrects for counting each diagonal twice.

Octagon check: a regular octagon with side 4 has perimeter 32, area 8 × 16 / (4 × tan(22.5°)) ≈ 77.25, interior angle 135°, exterior angle 45°, apothem ≈ 4.83, circumradius ≈ 5.23, and 20 diagonals.

Regular vs Irregular Polygons

Everything on this page assumes a regular polygon: all sides equal, all angles equal. If the sides or angles differ, one measurement cannot recover the shape — an irregular pentagon needs more side lengths, angles, coordinates, or a triangulation to pin down its area.

Checking Your Results

Verify any regular polygon answer with these quick audits:

• Perimeter must equal n × s exactly.
• Exterior angle must equal 360° / n, and interior + exterior must equal 180°.
• Diagonal count must equal n(n − 3)/2 (hexagon: 9, octagon: 20).
• Apothem must be smaller than the circumradius for any polygon with more than 3 sides.

Common Mistakes

• Degrees vs radians: the area and apothem formulas use tan(π/n) with the angle in radians, or tan(180°/n) in degrees. Mixing the two breaks the calculation.
• Confusing apothem with circumradius: apothem goes to the middle of a side, circumradius to a vertex.
• Using n = 2 or fractional sides: a polygon needs at least 3 sides, and n must be a whole number.

Polygon Names Reference

Apothem and Circumradius Relationship

The two radii are linked through the half central angle:

Check with the hexagon: R = 5, n = 6, so apothem = 5 × cos(30°) = 5 × 0.8660 ≈ 4.33 — matching the value computed from the side length.

Polygons in the Real World

• Honeycombs: hexagons tile a plane with minimal wax — nature's area-to-perimeter optimization.
• Nuts and bolts: hexagonal heads give a wrench six grip positions.
• Stop signs: regular octagons — eight equal sides are recognizable at a glance.
• Soccer balls: pentagons stitched with hexagons approximate a sphere.

As n Grows: Approaching a Circle

More sides means the polygon hugs its circumscribed circle more tightly. A 100-sided polygon with circumradius 1 has area (100/2) × sin(2π/100) ≈ 3.1395 — already within a tenth of a percent of π. In the limit, the polygon area formula becomes the circle's: A = πR².