Rhombus

Rhombus (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

Area of a Rhombus formulas
$A_{area} \;=\; \dfrac{ D' \cdot d' }{ 2 }$ $A_{area} \;=\; h \cdot a$ $A_{area} \;=\; a^2 \cdot sin( x)$ $A_{area} \;=\; 2\cdot a\cdot r$ $A_{area} \;=\; \dfrac{ 4\cdot r^2 }{ sin( x) }$
Symbol	English	Metric
$A_{area}$ = area	$in^2$	$mm^2$
$x$ = acute angles	$deg$	$rad$
$d', D'$ = diagonal	$in$	$mm$
$a, b, c, d$ = edge	$in$	$mm$
$r$ = inside radius	$in$	$mm$
$y$ = obtuce angles	$deg$	$rad$

Angle of a Rhombus formulas
$sin( x) \;=\; \dfrac{2\cdot D\cdot d '}{ D'^2 + d'^2}$ $sin( y) \;=\; \dfrac{2\cdot D' \cdot d'}{D'^2 + d'^2}$ $cos( x) \;=\; 1 - \dfrac{d'^2}{2\cdot a^2}$ $cos( x) \;=\; \dfrac{D'^2}{2\cdot a^2} - 1$ $cos( y) \;=\; 1 - \dfrac{D'^2}{2\cdot a^2}$ $cos( y) \;=\; \dfrac{d'^2}{2\cdot a^2} - 1$ $sin( x) \;=\; \dfrac{A}{a^2 }$ $sin( y) \;=\; \dfrac{A}{a^2 }$
Symbol	English	Metric
$x$ = acute angles	$deg$	$rad$
$A_{area}$ = area	$in^2$	$mm^2$
$d', D'$ = diagonal	$in$	$mm$
$a, b, c, d$ = edge	$in$	$mm$
$y$ = obtuce angles	$deg$	$rad$

Diagonal of a Rhombus formulas
$d' \;=\; \dfrac{2\cdot A_{area}}{D'}$ $D' \;=\; \dfrac {2\cdot A_{area}}{d'}$ $d' \;=\; \sqrt{ 4\cdot a^2 - D'^2 }$ $D' \;=\; \sqrt{ 4\cdot a^2 - d'^2 }$ $d' \;=\; a \cdot \sqrt{ 2 - 2 \cdot cos( x) }$ $d' \;=\; a \cdot \sqrt{ 2+ 2 \cdot cos( y) }$ $D' \;=\; a \cdot \sqrt{ 2 - 2 \cdot cos( y) }$ $D' \;=\; a \cdot \sqrt{ 2 + 2 \cdot cos( x) }$ $d' \;=\; 2\cdot a \cdot cos \left( \dfrac{y}{ 2} \right)$ $d' \;=\; 2\cdot a \cdot sin \left( \dfrac{x}{ 2} \right)$ $D' \;=\; 2\cdot a \cdot cos \left( \dfrac{x}{ 2} \right)$ $D' \;=\; 2\cdot a \cdot sin \left( \dfrac{y}{ 2} \right)$
Symbol	English	Metric
$d', D'$ = diagonal	$in$	$mm$
$x$ = acute angles	$deg$	$rad$
$A_{area}$ = area	$in^2$	$mm^2$
$a, b, c, d$ = edge	$in$	$mm$
$y$ = obtuce angles	$deg$	$rad$

Edge of a Rhombus formulas
$a \;=\; \dfrac{P}{4}$ $a \;=\; \dfrac{ \sqrt{ D'^2 + D'^2 } }{ 2 }$ $a \;=\; \sqrt{ \dfrac{ A }{ sin( x) } }$ $a \;=\; \sqrt{ \dfrac{ A }{ sin( y) } }$ $a \;=\; \dfrac{ d' }{ \sqrt{ 2 - 2 \cdot cos( x) } }$ $a \;=\; \dfrac{ d' }{ \sqrt{ 2 + 2 \cdot cos( y) } }$ $a \;=\; \dfrac{ D' }{ \sqrt{ 2 - 2 \cdot cos( y) } }$ $a \;=\; \dfrac{ D' }{ \sqrt{ 2 + 2 \cdot cos( x) } }$ $a \;=\; \sqrt{ \dfrac{ D' \cdot d' }{ 2 \cdot sin( x) } }$ $a \;=\; \sqrt{ \dfrac{ D' \cdot d' }{ 2 \cdot sin( y) } }$
Symbol	English	Metric
$a, b, c, d$ = edge	$in$	$mm$
$x$ = acute angles	$deg$	$\large{ rad}$
$A_{area}$ = area	$in^2$	$mm^2$
$d',\; D'$ = diagonal	$in$	$mm$
$P$ = perimeter	$in$	$mm$
$y$ = obtuce angles	$deg$	$rad$

Inscribed Circle Radius of a Rhombus formulas
$r \;=\; \dfrac{h}{2}$ $r \;=\; \dfrac{A_{area}}{2 a}$ $r \;=\; \dfrac{D' \cdot d'}{4 \cdot a}$ $r \;=\; \dfrac{ \sqrt{A_{area}\cdot sin( x) } }{2}$ $r \;=\; \dfrac{a\cdot sin( x) }{2}$ $r \;=\; \dfrac{a\cdot sin( y) }{2}$ $r \;=\; \dfrac{ D'\cdot sin \left(\dfrac{x}{2}\right) }{2}$ $r \;=\; \dfrac{ d'\cdot sin \left(\frac{y}{2}\right) }{2}$ $r \;=\; \dfrac{ D'\cdot d' }{ 2\cdot \sqrt{ D'^2 + d'^2 } }$
Symbol	English	Metric
$r$ = inside radius	$in$	$mm$
$A_{area}$ = area	$in^2$	$mm^2$
$d',\; D'$ = diagonal	$in$	$mm$
$a, b, c, d$ = edge	$in$	$mm$
$h$ = hight	$in$	$mm$

Perimeter of a Rhombus formulas
$P \;=\; 4\cdot a$
Symbol	English	Metric
$P$ = perimeter	$in$	$mm$
$a, b, c, d$ = edge	$in$	$mm$