Regular Polygon Perimeter Calculator

Regular Polygon Perimeter: Formula, Derivation, and Applications

What Is a Regular Polygon?

A regular polygon is a closed, two-dimensional shape in which every side is equal in length and every interior angle is equal in measure. Familiar examples include the equilateral triangle (3 sides, 60° angles), the square (4 sides, 90° angles), the regular pentagon (5 sides, 108° angles), the regular hexagon (6 sides, 120° angles), and the regular octagon (8 sides, 135° angles) — the shape displayed on stop signs worldwide. The perimeter of any regular polygon is the total length of its outer boundary, measured by summing all side lengths.

The Core Formula: P = n × s

The perimeter P of a regular polygon is calculated with a direct multiplication:

• P — Total perimeter in the chosen unit of length
• n — Number of sides (integer, minimum 3)
• s — Length of one side

Because every side in a regular polygon is congruent, multiplying the side count by a single side length yields the complete perimeter. A regular hexagon with each side measuring 5 cm produces P = 6 × 5 = 30 cm. A regular decagon (10 sides) with 3-inch sides gives P = 10 × 3 = 30 inches. The formula scales to any regular polygon regardless of size, as confirmed by the perimeter and area reference materials published by Texas Tech University's GEOM 1B Geometry I course.

Deriving the Side Length from the Apothem

The apothem (a) is the perpendicular distance from the polygon's center to the midpoint of any side — always shorter than the circumradius. When only the apothem is known, the side length is recovered using basic trigonometry:

• s = 2a × tan(π/n)
• P = n × s = 2n × a × tan(π/n)

For a regular hexagon (n = 6) with apothem 4 cm: s = 2 × 4 × tan(30°) = 8 × 0.5774 ≈ 4.619 cm, giving P = 6 × 4.619 ≈ 27.71 cm. For a square (n = 4) with apothem 5 cm: s = 2 × 5 × tan(45°) = 10 cm, so P = 4 × 10 = 40 cm — confirming the standard square perimeter formula.

Deriving the Side Length from the Circumradius

The circumradius (R) is the distance from the polygon's center to any vertex, always longer than the apothem. Given R, the side length follows from the sine rule for regular polygons:

• s = 2R × sin(π/n)
• P = n × s = 2n × R × sin(π/n)

A regular pentagon (n = 5) with circumradius 10 m: s = 2 × 10 × sin(36°) = 20 × 0.5878 ≈ 11.756 m, so P ≈ 58.78 m. An equilateral triangle (n = 3) with circumradius 10 cm: s = 2 × 10 × sin(60°) ≈ 17.32 cm, giving P ≈ 51.96 cm. These trigonometric relationships are implemented in professional engineering tools, as documented in the HP 35s Scientific Calculator Regular Polygon Equations reference.

Variable Reference

• n (Number of Sides): Integer ≥ 3. As n increases toward infinity, the polygon approaches a circle with circumference 2πR.
• s (Side Length): Direct edge measurement in any consistent unit (mm, cm, m, in, ft, yd).
• a (Apothem): Equals R × cos(π/n); always less than R.
• R (Circumradius): Equals a / cos(π/n); always greater than a.

Real-World Applications

Regular polygon perimeter calculations arise across architecture, manufacturing, and everyday design:

• Stop signs: Standard US stop signs are regular octagons with side lengths ≈ 12.5 inches; perimeter = 8 × 12.5 = 100 inches (≈ 254 cm).
• Hexagonal floor tiles: A 20 cm hexagonal tile has perimeter 6 × 20 = 120 cm, used to estimate grout and edge-trim quantities.
• Bolt heads: Hex bolt heads approximate regular hexagons; perimeter calculations inform wrench sizing and mechanical tolerances.
• Honeycomb cells: Natural honeycomb cells are regular hexagons ≈ 5.4 mm per side, giving a perimeter of roughly 32.4 mm per cell.
• Garden edging: Landscapers calculate regular polygon perimeters to estimate fencing and border materials for geometric planting beds.

Choosing the Right Input Mode

This regular polygon perimeter calculator accepts three measurement types and applies the appropriate formula automatically:

• Side Length: Direct computation — P = n × s.
• Apothem: Converts using s = 2a × tan(π/n), then applies P = n × s.
• Circumradius: Converts using s = 2R × sin(π/n), then applies P = n × s.

Select the input mode matching the available measurement to eliminate manual conversion errors and obtain an accurate perimeter result immediately.