Rectangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • rectangle 8Rectangle (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

Structural Shapes

Area of a Rectangle formula

\( \large{ A_{area} = a\;b  }\)

Where:

\(\large{ A_{area} }\) = area

\(\large{ a, b, c, d }\) = edge

Circumcircle Radius of a Rectangle formula

\( \large{ R =  \frac{D'}{2}    }\)

\( \large{ R =  \frac{  \sqrt{ a^2 \;+\; b^2 }  }{ 2 }   }\)

Where:

\(\large{ R }\) = outside radius

\(\large{ D' }\) = diagonal

\(\large{ a, b, c, d }\) = edge

Diagonal of a Rectangle formula

\( \large{ D' = \sqrt {a^2 + b^2 }   }\)

Where:

\(\large{ D' }\) = diagonal

\(\large{ a, b, c, d }\) = edge

Distance from Centroid of a Rectangle formula

\(\large{ C_x =  \frac{ b }{ 2 }  }\)

\(\large{ C_y =  \frac{ a }{ 2}  }\)

Where:

\(\large{ C }\) = distance from centroid

\(\large{ a, b, c, d }\) = edge

Elastic Section Modulus of a Rectangle formula

\( \large{ S_x =  \frac{ a^2\; b }{ 6  }  }\)

\( \large{ S_y =  \frac{ a\;b^2 }{ 6  }  }\)

Where:

\(\large{ S }\) = elastic section modulus

\(\large{ a, b, c, d }\) = edge

Perimeter of a Rectangle formula

\( \large{ P= 2\;a + 2\;b  }\)

\( \large{ P= 2\; \left( a + b     \right)  }\)

Where:

\(\large{ P }\) = perimeter

\(\large{ a, b, c, d }\) = edge

Plastic Section Modulus of a Rectangle formula

\( \large{ Z_x =  \frac{ a^2 \;b }{ 4  }   }\)

\( \large{ Z_y =  \frac{ a\;b^2 }{ 4  }   }\)

Where:

\(\large{ Z }\) = plastic section modulus

\(\large{ a, b, c, d }\) = edge

Polar Moment of Inertia of a Rectangle formula

\(\large{ J_{z} = \frac{a\;b}{12} \; \left( a^2 + b^2  \right)  }\)

\(\large{ J_{z1} = \frac{a\;b}{3} \; \left( a^2 + b^2  \right)  }\)

Where:

\(\large{ J }\) = torsional constant

\(\large{ a, b, c, d }\) = edge

Radius of Gyration of a Rectangle formula

\(\large{ k_{x} =    \frac{ a }{  2 \; \sqrt{3}  }    }\)

\(\large{ k_{y} =   \frac{ b }{  2 \; \sqrt{3}  }  }\)

\(\large{ k_{z} =   \sqrt{    \frac{ a^2 \;+\; b^2 }{ 2\; \sqrt{3} } } }\)

\(\large{ k_{x1} =   \frac{ a }  {  \sqrt 3  }  }\)

\(\large{ k_{y1} =  \frac{ b }  {  \sqrt 3  }  }\)

\(\large{ k_{z1} =  \sqrt{  \frac{ a^2 \;+\; b^2 }{  \sqrt{3} }  }  }\)

Where:

\(\large{ k }\) = radius of gyration

\(\large{ a, b, c, d }\) = edge

Second Moment of Area of a Rectangle formula

\(\large{ I_{x} =  \frac{a^3\; b}{12}  }\)

\(\large{ I_{y} = \frac{a\;b^3}{12}  }\)

\(\large{ I_{x1} =   \frac{a^3 \;b}{3}  }\)

\(\large{ I_{y1} =  \frac{a\;b^3}{3}  }\)

Where:

\(\large{ I }\) = moment of inertia

\(\large{ a, b, c, d }\) = edge

Side of a Rectangle formula

\( \large{ a = \frac{P}{2} \;-\; b   }\)

\( \large{ b = \frac{P}{2} \;-\; a  }\)

Where:

\(\large{ a, b, c, d }\) = edge

\(\large{ P }\) = perimeter

Torsional Constant of a Rectangle formula

\(\large{ J  =  a^3\; b\; \left[  \frac{1}{3} \;-\; \frac{0.21\;a}{b} \; \left(  1 \;-\; \frac{ a^4 }{  12\;b^4 } \right)  \right]    }\)

Where:

\(\large{ J }\) = torsional constant

\(\large{ a, b, c, d }\) = edge

 

Tags: Equations for Moment of Inertia Equations for Structural Steel Equations for Modulus