# Rhombus

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• 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

### 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