Regular Polygon

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • regular polygon 2Regular polygon (a two-dimensional figure) is a polygon where all sides are congruent and all angles are congruent.
  • Circumcircle is a circle that passes through all the vertices of a two-dimensional figure.
  • Congruent is all sides having the same lengths and angles measure the same.
  • Inscribed circle is the largest circle possible that can fit on the inside of a two-dimensional figure.
  • Polygon (a two-dimensional figure) is a closed plane figure for which all sides are line segments and not necessarly congruent.

Structural Shapes

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

area of a Regular Polygon formula

\(\large{ A_{area} =  \frac{a^2 \; n}{4 \; tan  \left( \frac{180}{n}  \right)  }  }\)

\(\large{ A_{area} =  \frac{R^2 \; n \; sin  \left( \frac{360}{n}  \right)   }{2}  }\)

\(\large{ A_{area} =  r^2 \; n \; tan  \left( \frac{180}{n}  \right)  }\)

\(\large{ A_{area} =  \frac{1}{4} \; a^2 \; n \; cot  \left( \frac{\pi}{n}  \right)  }\)

Where:

\(\large{ A_{area} }\) = area

\(\large{ a }\) = edge

\(\large{ r }\) = inside radius (apothem)

\(\large{ n }\) = number of edges

\(\large{ R }\) = outside radius

\(\large{ P }\) = perimeter

Central Angle of a Regular Polygon formula

\(\large{ CA = \frac{360}{n}   }\)

Where:

\(\large{ CA }\) = central angle

\(\large{ n }\) = number of edges

Circumcircle Radius of a Regular Polygon formula

\(\large{ R =  \frac{a}{2 \; sin \; \left( \frac{180}{n}  \right)  }  }\)

Where:

\(\large{ R }\) = outside radius

\(\large{ a }\) = edge

\(\large{ n }\) = number of edges

Distance from Centroid of a Polygon formula

\(\large{ C_x =  R  }\)

\(\large{ C_y =  R  }\)

Where:

\(\large{ C }\) = distance from centroid

\(\large{ R }\) = outside radius

Edge of a Regular Polygon formula

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

\(\large{ a =  2 \; R \; sin  \left( \frac{180}{n}  \right)   }\)

Where:

\(\large{ a }\) = edge

\(\large{ r }\) = inside radius (apothem)

\(\large{ R }\) = outside radius

Elastic Section Modulus of a Polygon formula

\( \large{ S =  \frac{ I_x }{ R }  }\)

Where:

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

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

\(\large{ R }\) = outside radius

Inscribed Radius of a Regular Polygon formula

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

\(\large{  r =  R \; cos  \left( \frac{180}{n}  \right)   }\)

Where:

\(\large{ r }\) = inside radius (apothem)

\(\large{ a }\) = edge

\(\large{ n }\) = number of edges

\(\large{ R }\) = outside radius

Number of Diagonals of a Regular Polygon formula

\(\large{ D' = \frac{ n \; \left( n \;-\; 3  \right)   }{2}   }\)

Where:

\(\large{ D' }\) = diagonal

\(\large{ a }\) = edge

\(\large{ n }\) = number of edges

Perimeter of a Regular Polygon formula

\(\large{ P = a \; n   }\)

Where:

\(\large{ P }\) = perimeter

\(\large{ a }\) = edge

\(\large{ n }\) = number of edges

Polar Moment of Inertia of a Polygon formula

\(\large{ J_{z} =  2 \; A  \left( \frac{ 6 \; R^2 \;-\; a^2 }{24}  \right)  }\)

Where:

\(\large{ J }\) = torsional constant

\(\large{ a }\) = edge

\(\large{ R }\) = outside radius

Radius of Gyration of a Polygon formula

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

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

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

Where:

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

\(\large{ a }\) = edge

\(\large{ R }\) = outside radius

Second Moment of Area of a Rectangle formula

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

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

Where:

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

\(\large{ a }\) = edge

\(\large{ R }\) = outside radius

 

Tags: Equations for Moment of Inertia Equations for Structural Steel Equations for Modulus