Regular Polygon

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.

Regular Polygon Index

Regular Polygon Types

  • 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 formulas

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

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

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

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

Symbol English Metric
\(\large{ A_{area} }\) = area  \(\large{ in^2 }\) \(\large{ mm^2 }\) 
\(\large{ a }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ r }\) = inside radius (apothem) \(\large{ in }\) \(\large{ mm }\)
\(\large{ n }\) = number of edges \(\large{ dimensionless }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{ mm }\)
\(\large{ P }\) = perimeter \(\large{ in }\) \(\large{ mm }\)

 

Central Angle of a Regular Polygon formulaCE

\(\large{ CA = \frac{360}{n}   }\) 
Symbol English Metric
\(\large{ CA }\) = central angle  \(\large{ deg }\) \(\large{ rad }\) 
\(\large{ n }\) = number of edges \(\large{ dimensionless }\)

 

Circumcircle Radius of a Regular Polygon formula

\(\large{ R =  \frac{a}{2 \; sin \; \left( \frac{180}{n}  \right)  }  }\)
Symbol English Metric
\(\large{ R }\) = outside radius  \(\large{ in }\) \(\large{ mm }\) 
\(\large{ a }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ n }\) = number of edges \(\large{ dimensionless }\)

 

Distance from Centroid of a Polygon formulas

\(\large{ C_x =  R  }\)

\(\large{ C_y =  R  }\)

Symbol English Metric
\(\large{ C }\) = distance from centroid \(\large{ in }\) \(\large{ mm }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{ mm }\)

 

Edge of a Regular Polygon formulas

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

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

Symbol English Metric
\(\large{ a }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ r }\) = inside radius (apothem) \(\large{ in }\) \(\large{ mm }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{ mm }\)

 

Elastic Section Modulus of a Polygon formula

\( \large{ S =  \frac{ I_x }{ R }  }\) 
Symbol English Metric
\(\large{ S }\) = elastic section modulus \(\large{ in^3 }\) \(\large{ mm^3 }\)
\(\large{ I }\) = moment of inertia \(\large{ in^4 }\) \(\large{ mm^4 }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{ mm }\)

 

Inscribed Radius of a Regular Polygon formulas

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

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

 

Symbol English Metric
\(\large{ r }\) = inside radius (apothem)  \(\large{ in }\) \(\large{ mm }\) 
\(\large{ a }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ n }\) = number of edges \(\large{ dimensionless }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{ mm }\)

 

Number of Diagonals of a Regular Polygon formula

\(\large{ D' = \frac{ n \; \left( n \;-\; 3  \right)   }{2}   }\) 
Symbol English Metric
\(\large{ D' }\) = diagonal  \(\large{ in }\) \(\large{ mm }\) 
\(\large{ a }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ n }\) = number of edges \(\large{ dimensionless }\)

   

Perimeter of a Regular Polygon formula

\(\large{ P = a \; n   }\) 
Symbol English Metric
\(\large{ P }\) = perimeter  \(\large{ in }\) \(\large{ mm }\) 
\(\large{ a }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ n }\) = number of edges \(\large{ dimensionless }\)

 

Polar Moment of Inertia of a Polygon formula

\(\large{ J_{z} =  2 \; A  \left( \frac{ 6 \; R^2 \;-\; a^2 }{24}  \right)  }\) 
Symbol English Metric
\(\large{ J }\) = torsional constant  \(\large{ in^4 }\) \(\large{ mm^4 }\) 
\(\large{ a }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{ mm }\)

 

Radius of Gyration of a Polygon formulas

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

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

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

Symbol English Metric
\(\large{ k }\) = radius of gyration  \(\large{ in }\) \(\large{ mm }\) 
\(\large{ a }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{ mm }\)

 

Second Moment of Area of a Rectangle formulas

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

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

Symbol English Metric
\(\large{ I }\) = moment of inertia  \(\large{ in^4 }\) \(\large{ mm^4 }\) 
\(\large{ a }\) = edge \(\large{ in }\) \(\large{ mm }\)
\(\large{ R }\) = outside radius \(\large{ in }\) \(\large{ mm }\)

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