Polygon

Written by Jerry Ratzlaff on . Posted in Plane Geometry

Polygon - Geometric Propertiespolygon 4A

A regular polygon (polyhedron) has equal side lengths and angle degrees.

Types of Regular Polygon

  • Triangle - 3 sides - 60° interior angle
  • Quadrilateral - 4 sides - 90° interior angle
  • Pentagon - 5 sides - 108° interior angle
  • Hexagon - 6 sides - 120° interior angle
  • Heptagon - 7 sides - 128.571° interior angle
  • Octagon - 8 sides - 135° interior angle
  • Nonagon - 9 sides - 140° interior angle
  • Decagon - 10 sides - 144° interior angle
  • Hendecagon - 11 sides - 147.273° interior angle
  • Dodecagon - 12 sides - 150° interior angle
  • Triskaidecagon - 13 sides - 152.308° interior angle
  • Tetrakaidecagon - 14 sides - 154.286° interior angle
  • Pentadecagon - 15 sides - 156° interior angle
  • Hexakaidecagon - 16 sides - 157.5° interior angle
  • Heptadecagon - 17 sides - 158.824° interior angle
  • Octakaidecagon - 18 sides - 160° interior angle
  • Enneadecagon - 19 sides - 161.053° interior angle
  • Icosagon - 20 sides - 162° interior angle

Angle of a Polygon formula

\(\large{  \theta =\frac { \pi \left (n-2 \right) } {n} = \pi - \phi  }\)

Apothem of a Polygon formula

\(\large{  a = \frac { s }{ 2 tan \; \left( \frac{180}{n}  \right)   }  }\)

area of a Polygon formula

\(\large{ A =  \frac{1}{2} P a   }\)

\(\large{ A =  \frac{Pa}{2}  }\)

\(\large{ A =  \frac{1}{4} s^2 \; n \; cos \; \left( \frac{ \pi }{n}   \right)     }\)

Central Angle of a Polygon formula

\(\large{ \phi =\frac {2 \pi}{ n} }\)

Perimeter of a Polygon formula

\(\large{ P = s n   }\)

Distance from Centroid of a Polygon formula

\(\large{ C_x =  r   }\)

\(\large{ C_y =  r   }\)

Elastic Section Modulus of a Polygon formula

\(\large{ S =  \frac{ I_x }{ r}  }\)

Polar Moment of Inertia of a Polygon formula

\(\large{ J_z =   2A  \left(  \frac{6r^2 \;-\; s^2}{24}  \right)   }\)

Radius of Gyration of a Polygon formula

\(\large{ k_{x} =   \sqrt { \frac{6r^2 \;-\; s^2 }{24}     }   }\)

\(\large{ k_{y} =  \sqrt { \frac{6r^2 \;-\; s^2 }{24}     }      }\)

\(\large{ k_{z} =  \sqrt { K_{x}{^2} \;+\; K_{y}{^2}    }       }\)

Second Moment of Area of a Polygon formula

\(\large{ I_{x} =  A  \left(  \frac{6r^2 \;-\; s^2}{24}  \right)  }\)

\(\large{ I_{y} =   A  \left(  \frac{6r^2 \;-\; s^2}{24}  \right)    }\)

Side Length of a Polygon formula

\(\large{ s = 2r \left[ tan \left(  \frac{\theta}{2} \right) \right]  }\)

 

Where:

\(\large{ a }\) = apothem

\(\large{ A }\) = area

\(\large{ C_x, C_y }\) = distance from centroid

\(\large{ d }\) = diameter

\(\large{ I }\) = moment of inertia

\(\large{ k }\) = radius of gyration

\(\large{ n }\) = number of flat sides

\(\large{ P }\) = perimeter

\(\large{ r }\) = radius

\(\large{ s }\) = side length

\(\large{ S }\) = elastic section modulus

\(\large{ \theta }\) = angle

\(\large{ \pi }\) = Pi

\(\large{ \phi }\) = central angle in radians

 

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