Regular Polygon

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • polygon 4AA regular polygon is a two-dimensional figure that is a polygon where all sides are congruent and all angles are congruent.
  • A polygon is a two-dimensional figure that is a closed plane figure for which all sides are line segments and not necessarly congruent.
  • A regular polygon is a structural shape used in construction.
  • See Geometric Properties of 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

Angle of a Regular Polygon formula

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

Where:

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

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

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

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

Apothem of a Regular Polygon formula

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

Where:

\(\large{ a }\) = apothem

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

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

area of a Regular Polygon formula

\(\large{ A =  \frac{1}{2}\; P\; a }\)

\(\large{ A =  \frac{P\;a}{2} }\)

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

Where:

\(\large{ A }\) = area

\(\large{ a }\) = apothem

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

\(\large{ P }\) = perimeter

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

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

Central Angle of a Regular Polygon formula

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

Where:

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

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

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

Perimeter of a Regular Polygon formula

\(\large{ P = s \;n   }\)

Where:

\(\large{ P }\) = perimeter

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

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

Distance from Centroid of a Polygon formula

\(\large{ C_x =  r  }\)

\(\large{ C_y =  r  }\)

Where:

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

\(\large{ r }\) = radius

Elastic Section Modulus of a Regular Polygon formula

\(\large{ S =  \frac{ I_x }{ r}  }\)

Where:

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

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

\(\large{ r }\) = radius

Polar Moment of Inertia of a Regular Polygon formula

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

Where:

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

\(\large{ A }\) = area

\(\large{ r }\) = radius

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

Radius of Gyration of a Regular Polygon formula

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

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

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

Where:

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

\(\large{ r }\) = radius

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

Second Moment of Area of a Regular Polygon formula

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

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

Where:

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

\(\large{ A }\) = area

\(\large{ r }\) = radius

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

Side Length of a Regular Polygon formula

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

Where:

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

\(\large{ r }\) = radius

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

 

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