Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • A two-dimensional figure that is a quadrilateral that has a pair of parallel opposite sides.
  • A trapezoid is a structural shape used in construction.
  • See Geometric Properties of Structural Shapes
  • No interior angles are equal.
  • Angle A & D = 180
  • Angle B & C = 180
  • 2 diagonals
  • 4 edges
  • 4 vertexs

trapezoid 1BArea of a Trapezoid formula

\(\large{  A_{area} =  h \; \left(  \frac  {a + b}  {2 }  \right)   }\)

\(\large{  A_{area} =  m\;h   }\)

Where:

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

\(\large{ h }\) = height

\(\large{ m }\) = midline

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

trapezoid 1BDiagonal of a Trapezoid Formula

\(\large{  d' = \sqrt{  b^2+c^2-2\;b \sqrt{c^2-h^2}  }  }\)

\(\large{  D' = \sqrt{  b^2+d^2-2\;b \sqrt{d^2-h^2}  }  }\)

Where:

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

\(\large{ h }\) = height

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

 

trapezoid 1BHeight of a Trapezoid formula

\(\large{  h =  \frac  { 2\; A_{area} }  { a + b }  }\)

\(\large{  h =  \frac  { A_{area} }  { m }  }\)

Where:

\(\large{ h }\) = height

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

\(\large{ h }\) = height

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

trapezoid 1BMidline of a Trapezoid formula

\(\large{  m = \frac{a + b}{2}   }\)

Where:

\(\large{ m }\) = midline

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

 

 

 

trapezoid 1BPerimeter of a Trapezoid formula

\(\large{  P =  a + b + c + d   }\)

\(\large{  P =  \sqrt {h^2 + g^2} + \sqrt {h^2 + \left( b \;-\; a \;-\; g  \right)^2   }  + a + b   }\)

Where:

\(\large{ P }\) = perimeter

\(\large{ h }\) = height

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

 

trapezoid 1BSide of a Trapezoid formula

\(\large{  a =  2 \; \frac { A_{area} }{h} \;-\; b   }\)

\(\large{  b =  2 \; \frac { A_{area} }{h} \;-\; a   }\)

\(\large{  c =  P \;-\; a \;-\; b \;-\; d   }\)

\(\large{  d =  P \;-\; a \;-\; b \;-\; c   }\)

Where:

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

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

\(\large{ h }\) = height

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

trapezoid 2ADistance from Centroid of a Trapezoid Formula

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

\(\large{  C_y =  \frac { h }  { 3}    \left(     \frac { 2\;a + b } { a + b }  \right)    }\)

Where:

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

\(\large{ h }\) = height

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

 

trapezoid 2AElastic Section Modulus of a Trapezoid formula

\(\large{  S_x =  \frac { I_x }  { C_y  }   }\)

\(\large{  S_y =  \frac { I_y }  { C_x  }   }\)

Where:

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

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

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

 

trapezoid 2APlastic Section Modulus of a Trapezoid formula

\(\large{  Z_x =  \frac {  h^2   \;  \left(  2\;a^2 + 14\;a\;b + 2\;b^2   \right)  }  { 12\; \left(  a + b   \right)  }   }\)

\(\large{  Z_y =  \frac {    6\;a\;b\;h \;-\;  3\;a^2\;h \;-\; 8\;a + 8\;b + 4\;g^2\;h  \;-\; 8\;g    }  {  24 }   }\)

Where:

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

\(\large{ h }\) = height

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

 

trapezoid 2APolar Moment of Inertia of a Trapezoid formula

\(\large{  J_{z} =  I_x \;+\; I_y    }\)

\(\large{  J_{z1} =  I_{x1}  +   I_{y1}    }\)

Where:

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

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

 

 

trapezoid 2ARadius of Gyration of a Trapezoid formula

\(\large{  k_{x} =    \frac { h }  { 6 }  \;   \sqrt  {   2 +  \frac  { 4\;a\;b}  { \left( a + b \right)^2 }  }      }\)

\(\large{  k_{y} =   \sqrt {  \frac {I_y} {A_{area}}    }    }\)

\(\large{  k_{z} =   \sqrt  { k_{x}{^2}  + k_{y}{^2}  }   }\)

\(\large{  k_{x1} =   \frac { 1 }  { 6 }     \sqrt  {  \frac  { 6\;h^2 \; \left( 3\;a + b \right)  }  {  a + b }   }    }\)

\(\large{  k_{y1} =    \sqrt {  \frac {I_{y1}} {A_{area}}    }    }\)

\(\large{  k_{z1} =  \sqrt  { k_{x1}{^2}  \;+\; k_{y1}{^2}  }    }\)

Where:

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

\(\large{ h }\) = height

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

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

trapezoid 2ASecond Moment of Area of a Trapezoid formula

\(\large{  I_{x} =  \frac {    h^3   \;  \left(  a^2 \;4\;a\;b + b^2   \right)    }  {  36   \;  \left(  a + b   \right) }   }\)

\(\large{  I_{y} = \frac {  h  \;  \left( 4\;a\;b\;g^2   + 3\;a^2\; b\;g  \;-\;  3\;a\;b^2 \;g   + a^4  + b^4  + 2\;a^3 \;b  + a^2 \;g^2  + a^3 \;g + 2\;a\;b^3  \;-\;  g\;b^3 +  b^2\;g^2    \right) }   { 36  \;   \left(  a + b   \right)  }   }\)

\(\large{  I_{x1} =   \frac {  h^3   \;  \left( 3a + b   \right)  }   { 12  }   }\)

\(\large{  I_{y1} =  \frac {  h  \;  \left( a^3 + 3\;a\;g^2  +  3\;a^2\;g + b^3  +  g\;b^2  + a\;b^2  + b\;g^2  + 2\;a\;b\;g + b\;a^2  \right)  }   {  12  }   }\)

Where:

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

\(\large{ h }\) = height

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

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

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

 

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