Regular Polygon Perimeter Calculator – Find the Perimeter Instantly
A regular polygon is a closed shape with all sides equal in length and all interior angles equal in measure. The perimeter of a regular polygon is simply the total distance around its boundary. This calculator lets you find the perimeter from any of three common measurements — the side length, the circumradius, or the apothem — and works for any polygon from a triangle right up to a 1000-gon.
Three Input Modes
Depending on which measurement you have available, the calculator uses one of three paths:
- Side Length (s): The direct formula. The perimeter is
P = n × s, where n is the number of sides and s is the length of one side. - Circumradius (R): The circumradius is the distance from the centre to any vertex (corner). The side length is first derived using
s = 2R × sin(π/n), then the perimeter is calculated asP = n × s. - Apothem (a): The apothem, also called the inradius, is the perpendicular distance from the centre to the midpoint of any side. The side is derived via
s = 2a × tan(π/n), and thenP = n × sgives the perimeter.
Understanding the Key Measurements
For any regular n-gon, three radial measurements describe the polygon completely:
- Side length (s): the length of each of the n equal edges.
- Circumradius (R): the radius of the circle that passes through all vertices. Always larger than the apothem.
R = s / (2 × sin(π/n)). - Apothem (a): the radius of the inscribed circle, tangent to the middle of each side.
a = s / (2 × tan(π/n)) = R × cos(π/n).
Angles of a Regular Polygon
The calculator also displays the interior and exterior angles, which are fixed for any given n:
- Interior angle:
θ = ((n − 2) × 180) / ndegrees. For a hexagon this is 120°, for a square 90°, for a triangle 60°. - Exterior angle:
360 / ndegrees. Interior and exterior angles always sum to 180° for any polygon. - Sum of interior angles:
(n − 2) × 180°. A triangle sums to 180°, a quadrilateral to 360°, and so on.
Named Polygon Presets
For the most common shapes, use the quick-preset buttons at the top of the calculator. These instantly set the number of sides to the correct value without typing:
- Triangle (3 sides) — equilateral when regular
- Square (4 sides)
- Pentagon (5 sides)
- Hexagon (6 sides) — used in honeycomb structures
- Heptagon (7 sides)
- Octagon (8 sides) — stop-sign shape
- Nonagon (9 sides)
- Decagon (10 sides)
Unit Support
The calculator supports seven length units: millimeters (mm), centimeters (cm), meters (m), kilometers (km), inches (in), feet (ft), and yards (yd). Internally all values are converted to meters for computation, then converted back to your chosen display unit. Switch units at any time and the result updates immediately.
Practical Applications
Regular polygon perimeter calculations appear in many real-world situations. Architects use them when designing rooms or buildings with polygonal floor plans. Engineers reference regular polygon geometry when designing gears, bolts (hexagonal heads), and pipe fittings. Landscape designers use them for gazebos, garden beds, and paved areas. Teachers and students use the formulas when studying plane geometry and trigonometry. The apothem-based input mode is especially useful when only the inscribed circle radius is known from a blueprint or technical drawing.
Diagonals of a Regular Polygon
As a bonus, the calculator shows the number of diagonals, computed as n(n − 3) / 2. A triangle has 0 diagonals, a square has 2, a pentagon has 5, a hexagon has 9, and a decagon has 35. This formula counts all line segments connecting non-adjacent vertices.
Tips for Accurate Results
For the most accurate perimeter calculation, ensure your input measurement is as precise as possible. Even small errors in the side length, circumradius, or apothem are multiplied by n when computing the perimeter, so high-sided polygons amplify measurement imprecision more than low-sided ones. Use 4–6 decimal places of input precision when working with large polygons or tight tolerances.