Rectangle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • rectangle 8Rectangle is a two-dimensional figure that is a quadrilateral with two pair of parallel edges.
  • Quadrilateral is a two-dimensional figure that is a polygon with four sides.
  • 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.
  • See Geometric Properties of Structural Shapes
  • a = c,  b = d
  • A = B = C = D = 360°
  • 4 interior angles are 90°
  • 2 diagonals
  • 4 edges
  • 4 vertexs

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

\(\large{ J  =  a^3 \;b\; \left[  \frac {16}{3} \;-\; \frac {3.36\;a}{b} \; \left(  1 \;-\; \frac {  a^4  } {  12\;b^4  } \right)    \right]    }\)

Where:

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

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

 

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