Isosceles Trapezoid

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • isosceles trapezoid 6Isosceles trapezoid (a two-dimensional figure) is a trapezoid with only one pair of parallel edges and having base angles that are the same.
  • Acute angle measures less than 90°.
  • 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.
  • Obtuse angle measures more than 90°.
  • a & c are bases
  • b & d are legs
  • a ∥ c
  • a ≠ c
  • b = d
  • ∠A & ∠D < 90°
  • ∠B & ∠C > 90°
  • ∠A = ∠D
  • ∠B = ∠C
  • ∠A + ∠B = 180°
  • ∠C + ∠D = 180°
  • ∠A + ∠C = 180°
  • ∠B + ∠D = 180°
  • 2 diagonals
  • 4 edges
  • 4 vertexs

Structural Shapes

Angle of a Isosceles Trapezoid formula

\(\large{  x = arccos \; \frac{\left(\frac{a \;-\; c}{2}\right)^2 \;+\; b^2 \;-\; h^2}{2\;\left(\frac{a \;-\; c}{2}\right)\;b} }\)

\(\large{  y =  180° - x  }\)

Where:

\(\large{ x }\) = acute angle

\(\large{ y }\) = obtuse angle

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

\(\large{ h }\) = height

Area of an Isosceles Trapezoid formula

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

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

\(\large{  A_{area} =  mc \; sin \; x  }\)

\(\large{  A_{area} =  mc \; sin \; y  }\)

Where:

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

\(\large{ m }\) = central median

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

\(\large{ h }\) = height

Circumcircle of an Isosceles Trapezoid formula

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

\(\large{  R =  \frac  { b\; D' \;a  }     {  4 \;\sqrt  { s \; \left( s \;-\; b \right) \; \left( s \;-\; D' \right) \left( s \;-\; a \right)   }  }    }\)          \(\large{  s =  \frac  {b \;+\; D' \;+\; a} {2}    }\)

\(\large{  R =  b \; \sqrt{   \frac{a\;c \;+\; b^2}{4\;b^2 \;-\; \left( a \;-\; c \right)^2}   }   }\)

Where:

\(\large{ R }\) = outside radius

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

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

Diagonal of an Isosceles Trapezoid Formula

\(\large{  D' = \sqrt { b^2  \;+\; c\;a   }   }\)

Where:

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

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

\(\large{ h }\) = height

Distance from Centroid of an Isosceles Trapezoid Formula

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

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

Where:

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

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

\(\large{ h }\) = height

Elastic Section Modulus of an Isosceles 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 an Isosceles Trapezoid formula

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

Where:

\(\large{ h }\) = height

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

Perimeter of an Isosceles Trapezoid formula

\(\large{  P = 2\;b + c + a   }\)

\(\large{  P = 2 \; \sqrt{ h^2 + \frac{\left(b \;-\; a\right)^2}{4} } + a + b   }\)

Where:

\(\large{ P }\) = perimeter

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

\(\large{ h }\) = height

Plastic Section Modulus of an Isosceles Trapezoid formula

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

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

Where:

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

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

\(\large{ h }\) = height

Polar Moment of Inertia of an Isosceles 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 an Isosceles Trapezoid formula

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

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

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

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

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

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

Where:

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

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

\(\large{ h }\) = height

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

Second Moment of Area of an Isosceles Trapezoid formula

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

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

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

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

Where:

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

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

\(\large{ h }\) = height

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

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

 

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