Regular Polygon Calculator

Regular Polygon
Choose a Regular Polygon:


Choose a Calculation:
a =  m

Units*:
Significant Figures:

Answer:

n = 5
a =
r =
R =
A =
P =
x =
y =

Pentagon Shape
(n=5)

Pentagon Diagram, 5 sided polygon with R = circumradius, r = inradius and a = side length

r = inradius (apothem)
R = circumradius
a = side length
n = number of sides
x = interior angle
y = exterior angle
A = area
P = perimeter
π = pi = 3.14159 ...
√ = square root

Polygon Calculator

Use this calculator to calculate properties of a regular polygon. Enter any 1 variable plus the number of sides or the polygon name. Calculates side length, inradius (apothem), circumradius, area and perimeter.  Calculate from an regular 3-gon up to a regular 1000-gon.

* Units: Note that units of length are shown for convenience.  They do not affect the calculations.  The units are in place to give an indication of the order of the calculated results such as ft, ft2 or ft3.  Any other base unit can be substituted.

Regular Polygon Formulas

A regular polygon is a polygon that is both equiangular and equilateral.  All sides are equal length placed around a common center so that all angles between sides are also equal.  When the number of sides, n, is equal to 3 it is an equilateral triangle and when n = 4 is is a square.

The following formulas were used to develop calculations for this calculator where a = side length, r = inradius (apothem), R = circumradius, A = area, P = perimeter, x = interior angle, y = exterior angle and n = number of sides.

  • Side Length a
    • a = 2r tan(π/n) = 2R sin(π/n)
  • Inradius r
    • r = (1/2)a cot(π/n) = R cos(π/n)
  • Circumradius R
    • R = (1/2) a csc(π/n) = r sec(π/n)
  • Area A
    • A = (1/4)na2 cot(π/n) = nr2 tan(π/n)
  • Perimeter P
    • P = na
  • Interior Angle x
    • x = ((n-2)π / n) radians = (((n-2)/n) x 180° ) degrees
  • Exterior Angle y
    • y = (2π / n) radians = (360° / n) degrees
    • polygon interior and exterior angles

Selected Polygons

Polygon Name n Polygon Shape x y
trigon
(equilateral triangle)
3 trigon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (1/3)π
= 60°
(2/3)π
= 120°
tetragon
(square)
4 tetragon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (2/4)π
= 90°
(2/4)π
= 90°
pentagon 5 pentagon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (3/5)π
= 108°
(2/5)π
= 72°
hexagon 6 hexagon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (4/6)π
= 120°
(2/6)π
= 60°
heptagon 7 heptagon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (5/7)π
= 900°/7
= 128.57°
(2/7)π
= 360°/7
= 51.43°
octagon 8 octagon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (6/8)π
= 135°
(2/8)π
= 45°
nonagon 9 nonagon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (7/9)π
= 140°
(2/9)π
= 40°
decagon 10 decagon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (8/10)π
= 144°
(2/10)π
= 36°
undecagon 11 undecagon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (9/11)π
= 1620°/11
= 147.27°
(2/11)π
= 360°/11
= 32.73°
dodecagon 12 dodecagon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (10/12)π
= 150°
(2/12)π
= 30°
tridecagon 13 tridecagon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (11/13)π
= 1980°/13
= 152.31°
(2/13)π
= 360°/13
= 27.69°
tetradecagon 14 tetradecagon diagram with inscribed and circumscribed circles, inradius, circumradius, side, interior angle and exterior angle (12/14)π
= 2160°/14
= 154.29°
(2/14)π
= 360°/14
= 25.71°

References

Zwillinger, Daniel (Editor-in-Chief). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, p. 323, 2003.

Weisstein, Eric W. "Regular Polygon." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RegularPolygon.htm

 

Cite this content, page or calculator as:

Furey, Edward "Regular Polygon Calculator" From http://www.CalculatorSoup.com - Online Calculator Resource.