Isosceles Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

 Isosceles Trapezoid - Geometric Propertiesisosceles trapezoid 4isosceles trapezoid 3

  • Trapezoid is a quadrilateral with only one pair of parallel edges.
  • Angle \(A = B\)
  • Angle \(C = D\)
  • 2 diagonals
  • 4 edges
  • 4 vertexs

Area of an Isosceles Trapezoid formula

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

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

Circumcircle of an Isosceles Trapezoid formula

The radius of a circumcircle (outer) of a square if given side of diagonal \(( R )\).

\(\large{  R =  \frac  { c D_1 a  }     {  4 \sqrt  {       s  \left( s - c \right)  \left( s - D_ 1 \right)  \left( s - a \right)   }  }   }\)          \(\large{ s =  \frac  {c+D_1+a} {2}    }\)

\(\large{  R =  \frac  { c D_1 b  }     {  4 \sqrt  { s  \left( s - c \right) \left( s - D_1 \right) \left( s - b \right)   }  }    }\)          \(\large{  s =  \frac  {c+D_1+b} {2}    }\)

Diagonal of an Isosceles Trapezoid Formula

\(\large{  D_1 = \sqrt { c^2  + ab   }   }\)

Height of an Isosceles Trapezoid formula

\(\large{  h =  \frac {1}{2}  \sqrt { 4c^3 -  {a+b} }   }\)

Perimeter of an Isosceles Trapezoid formula

\(\large{  P= 2c + a + b   }\)

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

Distance from Centroid of an Isosceles Trapezoid Formula

\(\large{  C_x =  \frac { b }  { 2 }    }\)

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

Elastic Section Modulus of an Isosceles Trapezoid formula

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

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

Plastic Section Modulus of an Isosceles Trapezoid formula

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

\(\large{  Z_y =  \frac {    h^2     \left(  11a^2 + 26ab + 11b^2   \right)    }  {  48     \left(  a + b   \right) }   }\)

Polar Moment of Inertia of an Isosceles Trapezoid formula

\(\large{  J_{z} =  I_x + I_y    }\)

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

Radius of Gyration of an Isosceles Trapezoid formula

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

\(\large{  k_{y} =   \frac { 1 }  { 12 }     \sqrt {  6  \left( a^2 + b^2 \right)  }    }\)

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

\(\large{  k_{x1} =   \frac { h }  { 6 }     \sqrt  {   6   \frac  { 3a + b}  {  a + b }   }    }\)

\(\large{  k_{y1} =    \sqrt  {  \frac  { 3b + 5a}  { 12 \left( b + a \right) } b   }    }\)

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

Second Moment of Area of an Isosceles Trapezoid formula

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

\(\large{  I_{y} = \frac {  h     \left( a + b   \right)  \left( a^2 + b^2   \right)  }   { 48  }   }\)

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

\(\large{  I_{y1} =  \frac {  h    \left( a + b   \right)  \left( a^2 + 7b^2   \right)  }   { 48  }   }\)

 

Where:

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

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

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

\(\large{ D_1 }\) = 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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