Regular Polygon

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 (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.

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

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	\( dimensionless \)	\( dimensionless \)

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	\( in \)	\( mm \)

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	\( in \)	\( mm \)

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	\( in \)	\( mm \)

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	\( in \)	\( mm \)

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	\( dimensionless \)	\( dimensionless \)

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	\( dimensionless \)	\( dimensionless \)

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	\( in \)	\( mm \)

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	\( in \)	\( mm \)

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	\( in \)	\( mm \)