Parallelogram

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • parallelogram 3Parallelogram (a two-dimensional figure) is a quadrilateral with two pairs of parallel opposite sides.
  • Acute angle measures less than 90°.
  • Diagonal is a line from one vertices to another that is non adjacent.
  • Obtuse angle measures more than 90°.
  • Opposite sides are congurent and parallel.
  • Polygon (a two-dimensional figure) is a closed plane figure for which all edges are line segments and not necessarly congruent.
  • Quadrilateral (a two-dimensional figure) is a polygon with four sides.
  • a ∥ c
  • b ∥ d
  • ∠A & ∠C < 90°
  • ∠B & ∠D > 90°
  • ∠A + ∠B = 180°
  • ∠C + ∠D = 180°
  • 2 diagonals
  • 4 edges
  • 4 vertexs

Angle of a Parallelogram formula

\(\large{ cos \; x = \frac {a^2 \;+\; b^2 \;-\; d'^2 }{2\;a\;b}   }\)

\(\large{ cos \; y = \frac {a^2 \;+\; b^2 \;-\; D'^2 }{2\;a\;b}   }\)

\(\large{ sin \; x =  sin \; y  \; \frac {A }{a\;b}   }\)

Where:

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

\(\large{ y }\) = obtuce angles

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

\(\large{ A, B, C, D }\) = vertex

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

Area of a Parallelogram formula

\(\large{ A_{area} = a\;h_a  }\)

\(\large{ A_{area} = b\;h_b  }\)

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

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

Where:

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

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

\(\large{ h_a, h_b }\) = height

Diagonal of a Parallelogram formula

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

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

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

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

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

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

Where:

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

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

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

\(\large{ y }\) = obtuce angles

Edge of a Parallelogram formula

\(\large{ a = \frac {P}{2} - b }\)

\(\large{ b = \frac {P}{2} - a }\)

\(\large{ b = \frac {A}{h} }\)

\(\large{ a = \frac {h_b}{sin\; x}  }\)

\(\large{ a = \frac {h_b}{sin\; y}  }\)

\(\large{ b = \frac {h_a}{sin\; x}  }\)

\(\large{ b = \frac {h_a}{sin\; y}  }\)

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

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

Where:

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

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

\(\large{ h_a, h_b }\) = height

\(\large{ P }\) = perimeter

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

\(\large{ y }\) = obtuce angles

Height of a Parallelogram formula

\(\large{ h_a = \frac {A}{b} }\)

\(\large{ h_a = b \; sin \; x }\)

\(\large{ h_a = b \; sin \; y }\)

\(\large{ h_b = a \; sin \; x }\)

\(\large{ h_b = a \; sin \; y }\)

Where:

\(\large{ h_a, h_b }\) = height

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

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

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

\(\large{ y }\) = obtuce angles

Perimeter of a Parallelogram formula

\(\large{ P = 2 \left( a+b \right) }\)

\(\large{ P = 2\;a + 2\;b }\)

\(\large{ P = 2\;a + \sqrt{ D'^2 + d'^2 - 4\;a^2 } }\)

\(\large{ P = 2\;b + \sqrt{ D'^2 + d'^2 - 4\;b^2 } }\)

Where:

\(\large{ P }\) = perimeter

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

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