## Parallelogram-Geometry Formulas

### Parallelogram

A parallelogram is a special type of quadrilateral that has equal and parallel opposite sides.

### The perimeter of a parallelogram:

1. Formula of parallelogram perimeter in terms of sides:

$P=2a+2b=2\left(a+b\right)$

2. Formula of parallelogram perimeter in terms of one side and diagonals:

$P=2a+\sqrt{2{d}_{1}^{2}+2{d}_{2}^{2}–4{a}^{2}}$
$P=2b+\sqrt{2{d}_{1}^{2}+2{d}_{2}^{2}–4{b}^{2}}$

3. Formula of parallelogram perimeter in terms of side, height and sine of an angle:

$P=2\left(b+\frac{{h}_{b}}{\mathrm{sin}\alpha }\right)$
$P=2\left(a+\frac{{h}_{a}}{\mathrm{sin}\alpha }\right)$

### The area of a parallelogram

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

$A=a·{h}_{a}$
$A=b·{h}_{b}$

2. Formula of parallelogram area in terms of sides and sine of an angle between this sides:

$A=ab\mathrm{sin}\alpha$
$A=ab\mathrm{sin}\beta$

3. Formula of parallelogram area in terms of diagonals and sine of an angle between diagonals:

$A=\frac{1}{2}{d}_{1}{d}_{2}\mathrm{sin}\gamma$
$A=\frac{1}{2}{d}_{1}{d}_{2}\mathrm{sin}\delta$

### Diagonal of a parallelogram:

1. Formula of parallelogram diagonal in terms of sides and cosine β (cosine theorem)

${d}_{1}=\sqrt{{a}^{2}+{b}^{2}–2ab·\mathrm{cos}\beta }$
${d}_{2}=\sqrt{{a}^{2}+{b}^{2}+2ab·\mathrm{cos}\beta }$

2. Formula of parallelogram diagonal in terms of sides and cosine α (cosine theorem)

${d}_{1}=\sqrt{{a}^{2}+{b}^{2}+2ab·\mathrm{cos}\alpha }$
${d}_{2}=\sqrt{{a}^{2}+{b}^{2}–2ab·\mathrm{cos}\alpha }$

3. Formula of parallelogram diagonal in terms of two sides and other diagonal:

${d}_{1}=\sqrt{2{a}^{2}+2{b}^{2}–{d}_{2}^{2}}$
${d}_{2}=\sqrt{2{a}^{2}+2{b}^{2}–{d}_{1}^{2}}$

4. Formula of parallelogram diagonal in terms of area, other diagonal and angles between diagonals:

${d}_{1}=\frac{2A}{{d}_{2}·\mathrm{sin}\gamma }=\frac{2A}{{d}_{2}·\mathrm{sin}\delta }$
${d}_{2}=\frac{2A}{{d}_{1}·\mathrm{sin}\gamma }=\frac{2A}{{d}_{1}·\mathrm{sin}\delta }$

### Characterizations of a parallelogram:

Quadrilateral ABCD is a parallelogram, if at least one of the following conditions:

• Quadrilateral has two pairs of parallel sides:

AB||CD, BC||AD

• Quadrilateral has a pair of parallel sides with equal lengths:

AB||CD, AB = CD (или BC||AD, BC = AD)

• Opposite sides are equal in the quadrilateral:

AB = CD, BC = AD

• Opposite angles are equal in the quadrilateral:

∠DAB = ∠BCD, ∠ABC = ∠CDA

• Diagonals bisect the intersection point in the quadrilateral:

AO = OC, BO = OD

• The sum of the quadrilateral angles adjacent to any side is 180°:

∠ABC + ∠BCD = ∠BCD + ∠CDA = ∠CDA + ∠DAB = ∠DAB + ∠DAB = 180°

• The sum of the diagonals squares equals the sum of the sides squares in the quadrilateral:

AC2 + BD2 = AB2 + BC2 + CD2 + AD2

### The basic properties of a parallelogram:

• Opposite sides of a parallelogram have the same length:

AB = CD, BC = AD

• Opposite sides of a parallelogram are parallel:

AB||CD, BC||AD

• Opposite angles of a parallelogram are equal:

∠ABC = ∠CDA, ∠BCD = ∠DAB

• Sum of the parallelogram angles is equal to 360°:

∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°

• The sum of the parallelogram angles adjacent to any sides is 180°:

∠ABC + ∠BCD = ∠BCD + ∠CDA = ∠CDA + ∠DAB = ∠DAB + ∠DAB = 180°

• Each diagonal divides the parallelogram into two equal triangle
• Two diagonals is divided parallelogram into two pairs of equal triangles
• The diagonals of a parallelogram intersect and intersection point separating each one in half:

AO = CO = d1/2

BO = DO = d2/2

• Intersection point of the diagonals is called a center of parallelogram symmetry
• Sum of the diagonals squares equals the sum of sides squares in parallelogram:

AC2 + BD2 = 2AB2 + 2BC2

• Bisectors of parallelogram opposite angles are always parallel
• Bisectors of parallelogram adjacent angles always intersect at right angles (90°)