Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • trapezoid 12Trapezoid (a two-dimensional figure) is a quadrilateral that has a pair of parallel opposite sides.
  • Acute angle measures less than 90°.
  • Diagonal is a line from one vertices to another that is non adjacent.
  • No interior angles are equal.
  • Obtuse angle measures more than 90°.
  • Quadrilateral (a two-dimensional figure) is a polygon with four sides.
  • See Geometric Properties of Structural Shapes
  • a & c are bases
  • b & d are legs
  • a ∥ c
  • a ≠ c
  • ∠A + ∠B = 180°
  • ∠C + ∠D = 180°
  • 2 diagonals
  • 4 edges
  • 4 vertexs

Area of a Trapezoid formula

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

\(\large{  A_{area} =  m\;h   }\)

Where:

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

\(\large{ h }\) = height

\(\large{ m }\) = midline

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

Diagonal of a Trapezoid Formula

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

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

Where:

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

\(\large{ h }\) = height

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

Distance from Centroid of a Trapezoid Formula

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

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

Where:

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

\(\large{ h }\) = height

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

Elastic 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

Height of a Trapezoid formula

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

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

Where:

\(\large{ h }\) = height

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

\(\large{ h }\) = height

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

Midline of a Trapezoid formula

\(\large{  m = \frac{a+c}{2}   }\)

Where:

\(\large{ m }\) = midline

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

Perimeter of a Trapezoid formula

\(\large{  P =  a + b + c + d   }\)

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

Where:

\(\large{ P }\) = perimeter

\(\large{ h }\) = height

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

Side of a Trapezoid formula

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

\(\large{  b =  P - c - a - d   }\)

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

\(\large{  d =  P - c - a - b   }\)

Where:

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

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

\(\large{ h }\) = height

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

Plastic Section Modulus of a Trapezoid formula

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

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

Where:

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

\(\large{ h }\) = height

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

Polar 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

Radius of Gyration of a Trapezoid formula

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

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

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

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

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

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

Where:

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

\(\large{ h }\) = height

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

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

Second Moment of Area of a Trapezoid formula

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

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

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

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

Where:

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

\(\large{ h }\) = height

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

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

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

 

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