Regular Polygon

on 27 January 2016. Posted in Plane Geometry

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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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 formula
\(\large{ CA = \frac{360}{n} }\)
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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