# Rhombus. Formulas, characterizations and properties of rhombus

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Definition of rhombus
Characterizations of rhombus
Basic properties of rhombus
Side of rhombus
Diagonals of rhombus
Perimeter of rhombus
Area of rhombus
Inradius of rhombus

Definition.

**Rhombus**is a parallelogram that has equal side. If the diamond angles is straight, then it is a square.

Rhombus differ in the size of side and the size of angles.

Fig.1 | Fig.2 |

## Characterizations of rhombus

Parallelogram

**ABCD**is a rhombus if at least one of the following conditions:1. Two consecutive sides are equal in length (it follows that all sides are the same):

АВ = ВС = СD = AD

2. If the diagonals are perpendicular:

AC┴BD

3. If the diagonals (angle bisectors) bisects an interior angles:

∠BAC = ∠CAD or ∠BDA = ∠BDC

4. If all the parallelogram heights have equal lenght:

BN = DL = BM = DK

5. If the parallelogram divided by diagonals into four equal rectangular triangles:

Δ ABO = Δ BCO = Δ CDO = Δ ADO

6. If in parallelogram can be inscribed circle.

## Basic properties of rhombus

1. It has all the properties of a parallelogram

2. A diagonals are perpendicular:

AC┴BD

3. A diagonals is angle bisectors:

∠BAC = ∠CAD, ∠ABD = ∠DBC, ∠BCA = ∠ACD, ∠ADB = ∠BDC

4. The sum of the squares of the diagonals equals the sum of the squares of the sides (the parallelogram law):

AC

^{2}+ BD

^{2}= 4AB

^{2}

5. The crosspoint of the diagonals is the center of symmetry.

6. At any rhombus can be inscribed circle.

7. The center of incircle is the diagonals crosspoint.

## Side of rhombus

### Side of rhombus formulas:

1. Formula of rhombus side in terms of area and height:

a = | S |

h_{a} |

2. Formulas of rhombus side in terms of area and sine of any angle:

a = | √S |

√sinα |

a = | √S |

√sinβ |

3. Formula of rhombus side in terms of area and inradius:

a = | S |

2r |

4. Formula of rhombus side in terms of it diagonals:

a = | √d_{1}^{2} + d_{2}^{2} |

2 |

5. Formulas of rhombus side in terms of diagonal and cosine of angle (cosα or cosβ):

a = | d_{1} |

√2 + 2 cosα |

a = | d_{2} |

√2 - 2 cosβ |

6. Formulas of rhombus side in terms of longer diagonal and half-angle:

a = | d_{1} |

2cos(α/2) |

a = | d_{1} |

2sin(β/2) |

7. Formulas of rhombus side in terms of smaller the diagonal and half-angle:

a = | d_{2} |

2cos(β/2) |

a = | d_{2} |

2sin(α/2) |

8. Formula of rhombus side in terms of perimeter:

a = | Р |

4 |

## Diagonals of rhombus

Definition.

**Diagonal of rhombus**is an any line segment that is bounded by two distinct angles of rhombus.

Rhombus has two diagonals the longer d

_{1}, and the smaller d_{2}### Diagonals of a rhombus formulas:

1. Formulas of rhombus longer diagonal in terms of side and cosine of any angle (cosα or cosβ)

d
d

_{1}= a√2 + 2 * cosα

_{1}= a√2 - 2 * cosβ

2. Formulas of rhombus smaller diagonal in terms of side and cosine of any angle (cosα or cosβ)

d
d

_{2}= a√2 + 2 * cosβ

_{2}= a√2 - 2 * cosα

3. Formulas of rhombus longer diagonal in terms of side and half-angle:

d
d

_{1}= 2a * cos(α/2)

_{1}= 2a * sin(β/2)

4. Formulas of rhombus smaller diagonal in terms of side and half-angle:

d
d

_{2}= 2a * sin(α/2)

_{2}= 2a * cos(β/2)

5. Formulas of rhombus diagonal in terms of side and other diagonal:

d
d

_{1}= √4a

^{2}- d

_{2}

^{2}

_{2}= √4a

^{2}- d

_{1}

^{2}

6. Formulas of rhombus diagonal in terms of other diagonal and tangents of half-angle:

d
d

_{1}= d

_{2}* tg(β/2)

_{2}= d

_{1}* tg(α/2)

7. Formulas of rhombus diagonal in terms of area and other diagonal:

d_{1} = | 2S |

d_{2} |

d_{2} = | 2S |

d_{1} |

8. Formulas of rhombus diagonals in terms of sine of half-angle and inradius:

d_{1} = | 2r |

sin(α/2) |

d_{2} = | 2r |

sin(β/2) |

## Perimeter of rhombus

Definition.

**Perimeter of rhombus**is the sum of all sides lenght of rhombus.

The lenght of side of rhombus can be find by the formulas listed above.

### Perimeter of a rhombus formula:

Formula of rhombus perimeter in terms of rhombus side:

P = 4a

## Area of rhombus

Definition.

**The area of rhombus**is the space limited by sides of rhombus, within the perimeter of rhombus.

### Area of a rhombus formulas:

1. Formula of rhombus area in terms of side and height:

S = a · h

_{a}

2. Formula of rhombus area in terms of side and sine any angles:

S = a

^{2}· sinα

3. Formula of rhombus area in terms of side and inradius:

S = 2a · r

4. Formula of rhombus area in terms of two diagonals:

S = | 1 | d_{1}d_{2} |

2 |

5. Formula of rhombus area in terms of sine angles and inradius:

S = | 4r^{2} |

sinα |

6. Formulas of rhombus area in terms of diagonal and the tangents half-angle:

S = | 1 | d_{1}^{2} · tg(α/2) |

2 |

S = | 1 | d_{2}^{2} · tg(β/2) |

2 |

## Incircle of a rhombus

Definition.

**Incircle of a rhombus**is the largest circle contained in the rhombus and it touches the four sides of a rhombus. The center of the incircle is called the rhombus'es incenter and disposed on the crosspoint of diagonales.

### Inradius of a rhombus formulas:

1. Formula of rhombus inradius in terms of height:

r = | h |

2 |

2. Formula of rhombus inradius in terms of area and the side:

r = | S |

2a |

3. Formula of rhombus inradius in terms of area and sine angles:

r = | √S · sinα |

2 |

4. Formulas of rhombus inradius in terms of side and sine any angles:

r = | a · sinα |

2 |

r = | a · sinβ |

2 |

5. Formulas of rhombus inradius in terms of diagonal and sine half-angle:

r = | d_{1} · sin(α/2) |

2 |

r = | d_{2} · sin(β/2) |

2 |

6. Formula of rhombus inradius in terms of rhombus diagonals:

r = | d_{1} · d_{2} |

2√d_{1}^{2} + d_{2}^{2} |

7. Formula of rhombus inradius in terms of rhombus diagonals and side:

r = | d_{1} · d_{2} |

4a |

Geometry formulas
Square. Formulas and Properties of a Square
Rectangle. Formulas and Properties of a Rectangle
Parallelogram. Formulas and Properties of a Parallelogram.
Rhombus. Formulas and Properties of a Rhombus
Area Formulas for Geometric Figures
Perimeter Formulas for Geometric Figures
Volume Formulas for Geometric Figures
Surface Area Formulas for Geometric Figures

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