# Rhombus

Written by Jerry Ratzlaff on . Posted in Plane Geometry

## Rhombus

• Rhombus is a quadrilateral with two pair of parallel edges.
• All edges are equal in length and opposite angles are equal.
• 4 angle
• 2 diagonals
• 4 edges
• 4 vertexs

### Area of a Rhombus formula

$$\large{ A = \frac {D' \; d' } {2} }$$

### Angle of a Rhombus formula

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

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

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

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

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

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

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

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

### Diagonal of a Rhombus formula

$$\large{ d' = \frac {2A}{D'} }$$

$$\large{ D' = \frac {2A}{d'} }$$

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

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

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

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

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

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

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

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

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

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

### Edge of a Rhombus formulacos

$$\large{ a = \frac {P} {4} }$$

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

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

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

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

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

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

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

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

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

### Perimeter of a Rhombus formula

$$\large{ P = 4a }$$

Where:

$$\large{ A }$$ = area

$$\large{ a }$$ = edge

$$\large{ A,\; B,\; C,\; D }$$ = vertex

$$\large{ d',\; D' }$$ = diagonal

$$\large{ P }$$ = perimeter

$$\large{ x }$$ = acute angle

$$\large{ y }$$ = obtuse angle