Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

Trapezoid - Geometric Propertiestrapezoid 11trapezoid 10

  • Trapezoid is a quadrilateral with only one pair of parallel edges.
  • No interior angles are equal.
  • 2 diagonals
  • 4 edges
  • 4 vertexs

Area of a Trapezoid formula

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

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

Diagonal of a Trapezoid Formula

\(\large{  D_1 = \sqrt { b^2 + d^2 - 2 * b * d * cos\;B  }    }\)

\(\large{  D_1 = \sqrt { a^2 + c^2 - 2 * a * c * cos\;D  }    }\)

\(\large{  D_2 = \sqrt { b^2 + c^2 - 2 * b * c * cos\;A  }    }\)

\(\large{  D_2 = \sqrt { d^2 + a^2 - 2 * d * a * cos\;C  }    }\)

Height of a Trapezoid formula

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

\(\large{  h =  d * sin \; C   }\)

\(\large{  h =  c * sin \; D   }\)

Perimeter of a Trapezoid formula

\(\large{  P =  a + b + c + d   }\)

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

Side of a Trapezoid formula

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

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

\(\large{  c =  P - a - b - d   }\)

\(\large{  d =  P - a - b - c   }\)

Distance from Centroid of a Trapezoid Formula

\(\large{  C_x =  \frac {  2ag + a^2 + gb + ab + b^2  }  {  3   \left(  { a+ b }  \right)  }   }\)

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

Elastic Section Modulus of a Trapezoid formula

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

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

Plastic Section Modulus of a Trapezoid formula

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

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

Moment of Inertia about Axis of a Trapezoid formula

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

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

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

\(\large{  I_{y1} =  \frac {  h    \left( a^3 + 3ag^2  +  3a^2g + b^3  +  gb^2  + ab^2  + bg^2  + 2abg + ba^2  \right)  }   {  12  }   }\)

Polar Moment of Inertia about Axis of a Trapezoid formula

\(\large{  J_{z} =  I_x + I_y    }\)

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

Radius of Gyration about Axis of a Trapezoid formula

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

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

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

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

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

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

 

Where:

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

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

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

\(\large{ D_1, D_2 }\) = diagonal

\(\large{ h }\) = height

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

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

\(\large{ P }\) = perimeter

\(\large{ r }\) = incircle

\(\large{ R }\) = outcircle

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

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

 

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