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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(a+b) P = 2a + 2b = 2(a + b)

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

P=2a+2d12+2d22–4a2 P = 2a + \sqrt {2d_1^2 + 2d_2^2 – 4a^2}
P=2b+2d12+2d22–4b2 P = 2b + \sqrt {2d_1^2 + 2d_2^2 – 4b^2}

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

P=2b+hbsinα P=2 \left( b+ \frac {h_b}{\sin \alpha} \right)
P=2a+hasinα P=2 \left( a+ \frac {h_a}{\sin \alpha} \right)

The area of a parallelogram

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

A=a·ha A= a \cdot h_a
A=b·hb A= b \cdot h_b

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

A=absinα A = ab \sin \alpha
A=absinβ A = ab \sin \beta

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

A=12d1d2sinγ A = \frac 12 d_1d_2 \sin \gamma
A=12d1d2sinδ A = \frac 12 d_1d_2 \sin \delta

Diagonal of a parallelogram:

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

d1=a2+b2–2ab·cosβ d_1 = \sqrt {a^2 + b^2 – 2ab\cdot \cos \beta}
d2=a2+b2+2ab·cosβ d_2 = \sqrt {a^2 + b^2 + 2ab\cdot \cos \beta}

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

d1=a2+b2+2ab·cosα d_1 = \sqrt {a^2 + b^2 + 2ab\cdot \cos \alpha}
d2=a2+b2–2ab·cosα d_2 = \sqrt {a^2 + b^2 – 2ab\cdot \cos \alpha}

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

d1=2a2+2b2–d22 d_1 = \sqrt {2a^2 + 2b^2 – d_2^2 }
d2=2a2+2b2–d12 d_2 = \sqrt {2a^2 + 2b^2 – d_1^2 }

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

d1=2Ad2·sinγ=2Ad2·sinδ d_1 = \frac {2A}{d_2\cdot \sin \gamma}= \frac {2A}{d_2\cdot \sin \delta}
d2=2Ad1·sinγ=2Ad1·sinδ d_2 = \frac {2A}{d_1\cdot \sin \gamma}= \frac {2A}{d_1\cdot \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°)

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