Rhombus

Written by Jerry Ratzlaff on . Posted in Plane Geometry

  • rhombus 5Rhombus (a two-dimensional figure) is a parallelogram with four congruent sides.
  • Acute angle measures less than 90°.
  • Congruent is all sides having the same lengths and angles measure the same.
  • Diagonal is a line from one vertices to another that is non adjacent.
  • Inscribed circle is the largest circle possible that can fit on the inside of a two-dimensional figure.
  • Obtuse angle measures more than 90°.
  • Parallelogram (a two-dimensional figure) is a quadrilateral with two pairs of parallel opposite sides.
  • a ∥ c
  • b ∥ d
  • a = b = c = d
  • ∠A & ∠C < 90°
  • ∠B & ∠D > 90°
  • ∠A + ∠B = 180°
  • ∠C + ∠D = 180°
  • 4 angle
  • 2 diagonals
  • 4 edges
  • 4 vertexs

Angle of a Rhombus Formula

\(\large{  y =  180° - x }\)

Where:

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

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

Area of a Rhombus formula

\(\large{ A_{area} = \frac {D' \;d' } {2} }\)

\(\large{ A_{area} = h \;a }\)

\(\large{ A_{area} = a^2 \; sin\; x }\)

\(\large{ A_{area} = 2\; a\; r }\)

\(\large{ A_{area} = \frac{4\; r^2}{sin\;x} }\)

Where:

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

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

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

\(\large{ r }\) = inside radius

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

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

Angle of a Rhombus formula

\(\large{ sin \; x =  \frac {2\;D'\;d'}{D'^2 \;+\; d'^2}  }\)

\(\large{ sin \; y =  \frac {2\;D'\;d'}{D'^2 \;+\; d'^2}  }\)

\(\large{ cos \; x = 1 - \frac {d'^2}{2\; a^2}  }\)

\(\large{ cos \; x = \frac {D'^2}{2\; a^2} - 1 }\)

\(\large{ cos \; y = 1 - \frac {D'^2}{2\; a^2}  }\)

\(\large{ cos \; y = \frac {d'^2}{2\; a^2} - 1 }\)

\(\large{ sin \; x =  \frac {A}{a^2 }  }\)

\(\large{ sin \; y =  \frac {A}{a^2 }  }\)

Where:

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

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

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

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

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

Diagonal of a Rhombus formula

\(\large{ d' = \frac {2\;A_{area}}{D'} }\)

\(\large{ D' = \frac {2\;A_{area}}{d'} }\)

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

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

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

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

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

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

\(\large{ d' = 2\;a \; cos \left(  \frac{y}{ 2} \right)    }\)

\(\large{ d' = 2\;a \; sin \left(  \frac{x}{ 2} \right)    }\)

\(\large{ D' = 2\;a \; cos \left(  \frac{x}{ 2} \right)    }\)

\(\large{ D' = 2\;a \; sin \left(  \frac{y}{ 2} \right)    }\)

Where:

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

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

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

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

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

Edge of a Rhombus formulacos

\(\large{ a = \frac {P} {4} }\)

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

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

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

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

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

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

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

\(\large{ a =   \sqrt{  \frac{ D' \; d' }{ 2 \; sin \; x }  }   }\)

 \(\large{ a =   \sqrt{  \frac{ D' \; d' }{ 2 \; sin \; y }  }   }\)

Where:

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

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

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

\(\large{ P }\) = perimeter

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

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

Inscribed Circle Radius of a Rhombus formula

\(\large{ r = \frac{h}{2}   }\)

\(\large{ r = \frac{A_{area}}{2 a}   }\)

\(\large{ r = \frac{D' \; d'}{4 a}   }\)

\(\large{ r = \frac{ \sqrt{A_{area}\;sin\;x } }{2}   }\)

\(\large{ r = \frac{a\;sin\;x}{2}   }\)

\(\large{ r = \frac{a\;sin\;y}{2}   }\)

\(\large{ r = \frac{ D'\;sin \frac{x}{2}  }{2}   }\)

\(\large{ r = \frac{ d'\;sin \frac{y}{2}  }{2}   }\)

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

Where:

\(\large{ r }\) = inside radius

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

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

\(\large{ h }\) = hight

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

Perimeter of a Rhombus formula

\(\large{ P = 4\;a }\)

Where:

\(\large{ P }\) = perimeter

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