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Regular Polygon

Regular Polygon Types

  • Triangle - 3 sides - 60° interior angl
  • 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

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

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

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

\( A_{area} \;=\; \dfrac{1}{4} \cdot a^2 \cdot n \cdot cot \left( \dfrac{\pi}{n} \right)  \)
Symbol English Metric
\( A_{area} \) = area  \( in^2 \) \( mm^2 \) 
\( a \) = edge \( in \) \( mm \)
\( r \) = inside radius (apothem) \( in \) \( mm \)
\( n \) = number of edges \( dimensionless \) \( dimensionless \)
\( R \) = outside radius \( in \) \( mm \)
\( P \) = perimeter \( in \) \( mm \)

regular polygon 2

  • Regular 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.

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Central Angle of a Regular Polygon formula
\( CA \;=\;  \dfrac{ 360}{n }\) 
Symbol English Metric
\( CA \) = central angle  \( deg \) \( rad \) 
\( n \) = number of edges \( dimensionless \) \( dimensionless \)

regular polygon 2 

 

 

 

 

Circumcircle Radius of a Regular Polygon formula
\( R \;=\;   \dfrac{ a }{ 2 \cdot sin \left( \dfrac{180}{n} \right)  }\)
Symbol English Metric
\( R \) = outside radius \( in \) \( mm \)
\( a \) = edge \( in \) \( mm \)
\( n \) = number of edges \( dimensionless \) \( dimensionless \)

regular polygon 2 

 

 

 

 

 

 

Distance from Centroid of a Polygon formulas

\( C_x \;=\; R  \)

\( C_y \;=\; R  \)
Symbol English Metric
\( C \) = distance from centroid \( in \) \( mm \)
\( R \) = outside radius \( in \) \( mm \)

regular polygon 2

 

 

 

 

 

Edge of a Regular Polygon formulas

\( a \;=\; 2 \cdot r \cdot tan \left( \dfrac{180}{n} \right)  \)

\( a \;=\; 2 \; R \cdot sin \left( \dfrac{180}{n} \right)   \)
Symbol English Metric
\( a \) = edge \( in \) \( mm \)
\( r \) = inside radius (apothem) \( in \) \( mm \)
\( R \) = outside radius \( in \) \( mm \)

regular polygon 2

 

 

 

 

 

 

 

Elastic Section Modulus of a Polygon formula
\( S \;=\;  \dfrac{ I_x }{ R }\) 
Symbol English Metric
\( S \) = elastic section modulus \( in^3 \) \( mm^3 \)
\( I \) = moment of inertia \( in^4 \) \( mm^4 \)
\( R \) = outside radius \( in \) \( mm \)

regular polygon 2 

 

 

 

 

 

 

Inscribed Radius of a Regular Polygon formulas

\(  r \;=\;  \dfrac{ a }{ 2\cdot tan \left( \dfrac{180}{n} \right)  }\) 

\(  r \;=\; R \cdot cos \left( \dfrac{180}{n} \right)   \)
Symbol English Metric
\( r \) = inside radius (apothem) \( in \) \( mm \)
\( a \) = edge \( in \) \( mm \)
\( n \) = number of edges \( dimensionless \) \( dimensionless \)
\( R \) = outside radius \( in \) \( mm \)

regular polygon 2 

 

 

 

 

 

 

 

 

Number of Diagonals of a Regular Polygon formula
\( D' \;=\;  \dfrac{ n \cdot ( n - 3 )  }{ 2 } \) 
Symbol English Metric
\( D' \) = diagonal \( in \) \( mm \)
\( a \) = edge \( in \) \( mm \)
\( n \) = number of edges \( dimensionless \) \( dimensionless \)

regular polygon 2   

 

 

 

 

 

 

Perimeter of a Regular Polygon formula
\( P \;=\;   a \cdot n  \) 
Symbol English Metric
\( P \) = perimeter \( in \) \( mm \)
\( a \) = edge \( in \) \( mm \)
\( n \) = number of edges \( dimensionless \) \( dimensionless \)

regular polygon 2 

 

 

 

 

 

Polar Moment of Inertia of a Polygon formula
\( J_{z} \;=\;   2 \cdot A \cdot  \dfrac{ 6 \cdot R^2 - a^2 }{ 24 } \) 
Symbol English Metric
\( J \) = torsional constant  \( in^4 \) \( mm^4 \) 
\( a \) = edge \( in \) \( mm \)
\( R \) = outside radius \( in \) \( mm \)

regular polygon 2 

 

 

 

 

 

 

Radius of Gyration of a Polygon formulas

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

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

\( k_{z} \;=\;   \sqrt{  k_{x}{^2}  +  k_{y}{^2}  }   \)
Symbol English Metric
\( k \) = radius of gyration \( in \) \( mm \)
\( a \) = edge \( in \) \( mm \)
\( R \) = outside radius \( in \) \( mm \)

regular polygon 2 

 

 

 

 

 

 

 

 

Second Moment of Area of a Rectangle formulas

\( I_{x} \;=\;   2 \cdot A \cdot  \dfrac{  6 \cdot R^2 - a^2 }{ 24 }  \) 

\( I_{y} \;=\;   2 \cdot A \cdot  \dfrac{  6 \cdot R^2 - a^2 }{ 24 }  \)
Symbol English Metric
\( I \) = moment of inertia  \( in^4 \) \( mm^4 \) 
\( a \) = edge \( in \) \( mm \)
\( R \) = outside radius \( in \) \( mm \)

regular polygon 2