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Regular Polygon Calculator

Polygon Calculator




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Pentagon Shape
n=5
A 5 sided polygon

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

Calculator Use

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)

a 3 sided polygon
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)

a 4 sided polygon
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

a 5 sided polygon
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

a 6 sided polygon
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

a 7 sided polygon
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

an 8 sided polygon
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

a 9 sided polygon
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

a 10 sided polygon
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

an 11 sided polygon
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

a 12 sided polygon
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

a 13 sided polygon
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

a 14 sided polygon
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. Regular Polygon.



 

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Furey, Edward "Regular Polygon Calculator"; from http://www.calculatorsoup.com - Online Calculator Resource.

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