Rectangle

on 29 January 2016. Posted in Plane Geometry

Rectangle (a two-dimensional figure) is a quadrilateral with two pair of parallel edges.
Circumcircle is a circle that passes through all the vertices of a two-dimensional figure.
Diagonal is a line from one vertices to another that is non adjacent.
Polygon (a two-dimensional figure) is a closed plane figure for which all edges are line segments and not necessarly congruent.
Quadrilateral (a two-dimensional figure) is a polygon with four sides.
a ∥ c
b ∥ d
a = c
b = d
∠A = ∠B = ∠C = ∠D = 360°
4 interior angles are 90°
2 diagonals
4 edges
4 vertexs

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Circumcircle Radius of a Rectangle formulas
\( \large{ R = \frac{D'}{2} }\) \( \large{ R = \frac{ \sqrt{ a^2 \;+\; b^2 } }{ 2 } }\)
Symbol	English	Metric
\(\large{ R }\) = outside radius	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ D' }\) = diagonal	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in }\)	\(\large{ mm }\)

Diagonal of a Rectangle formula
\( \large{ D' = \sqrt {a^2 + b^2 } }\)
Symbol	English	Metric
\(\large{ D' }\) = diagonal	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in }\)	\(\large{ mm }\)

Distance from Centroid of a Rectangle formulas
\(\large{ C_x = \frac{ b }{ 2 } }\) \(\large{ C_y = \frac{ a }{ 2} }\)
Symbol	English	Metric
\(\large{ C }\) = distance from centroid	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in }\)	\(\large{ mm }\)

Elastic Section Modulus of a Rectangle formulas
\( \large{ S_x = \frac{ a^2\; b }{ 6 } }\) \( \large{ S_y = \frac{ a\;b^2 }{ 6 } }\)
Symbol	English	Metric
\(\large{ S }\) = elastic section modulus	\(\large{in^3}\)	\(\large{mm^3}\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in }\)	\(\large{ mm }\)

Perimeter of a Rectangle formulas
\( \large{ P = 2\;a + 2\;b }\) \( \large{ P = 2\; \left( a + b \right) }\)
Symbol	English	Metric
\(\large{ P }\) = perimeter	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in }\)	\(\large{ mm }\)

Plastic Section Modulus of a Rectangle formulas
\( \large{ Z_x = \frac{ a^2 \;b }{ 4 } }\) \( \large{ Z_y = \frac{ a\;b^2 }{ 4 } }\)
Symbol	English	Metric
\(\large{ Z }\) = plastic section modulus	\(\large{ in^3 }\)	\(\large{ mm^3 }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in }\)	\(\large{ mm }\)

Polar Moment of Inertia of a Rectangle formulas
\(\large{ J_{z} = \frac{a\;b}{12} \; \left( a^2 + b^2 \right) }\) \(\large{ J_{z1} = \frac{a\;b}{3} \; \left( a^2 + b^2 \right) }\)
Symbol	English	Metric
\(\large{ J }\) = torsional constant	\(\large{ in^4 }\)	\(\large{ mm^4 }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in }\)	\(\large{ mm }\)

Radius of Gyration of a Rectangle formulas
\(\large{ k_{x} = \frac{ a }{ 2 \; \sqrt{3} } }\) \(\large{ k_{y} = \frac{ b }{ 2 \; \sqrt{3} } }\) \(\large{ k_{z} = \sqrt{ \frac{ a^2 \;+\; b^2 }{ 2\; \sqrt{3} } } }\) \(\large{ k_{x1} = \frac{ a } { \sqrt 3 } }\) \(\large{ k_{y1} = \frac{ b } { \sqrt 3 } }\) \(\large{ k_{z1} = \sqrt{ \frac{ a^2 \;+\; b^2 }{ \sqrt{3} } } }\)
Symbol	English	Metric
\(\large{ k }\) = radius of gyration	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in }\)	\(\large{ mm }\)

Second Moment of Area of a Rectangle formulas
\(\large{ I_{x} = \frac{a^3\; b}{12} }\) \(\large{ I_{y} = \frac{a\;b^3}{12} }\) \(\large{ I_{x1} = \frac{a^3 \;b}{3} }\) \(\large{ I_{y1} = \frac{a\;b^3}{3} }\)
Symbol	English	Metric
\(\large{ I }\) = moment of inertia	\(\large{in^4}\)	\(\large{mm^4}\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in }\)	\(\large{ mm }\)

Torsional Constant of a Rectangle formula
\(\large{ J = a^3\; b\; \left[ \frac{1}{3} \;-\; \frac{0.21\;a}{b} \; \left( 1 \;-\; \frac{ a^4 }{ 12\;b^4 } \right) \right] }\)
Symbol	English	Metric
\(\large{ J }\) = torsional constant	\(\large{ in^4 }\)	\(\large{ mm^4 }\)
\(\large{ a, b, c, d }\) = edge	\(\large{ in }\)	\(\large{ mm }\)

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