Parallelogram

A parallelogram is a quadrilateral where the opposite sides are equal in length and parallel.

The figure can be seen as a "slanted rectangle".

 

Parallelogram

 

There are no right angles in a parallelogram, but the opposite angles are always equal. The sum of two adjacent angles is 180°.

 

$$ \large \angle A + \angle B = 180^\circ $$

 

A special variation of the parallelogram is called a rhombus. Here all four sides are equal in length. In the same way as an equilateral rectangle is called a square.

 

The intersection point of the diagonals in a parallelogram is also their midpoint.

This means that the diagonals divide each other into two equal parts, but they are not necessarily perpendicular.

 

Properties of a parallelogram

  • Opposite sides are equal in length.
  • Opposite sides are parallel.
  • Opposite angles are equal.
  • Adjacent angles add up to 180°.
  • The intersection point of the diagonals is also their midpoint.
  • A parallelogram can be divided into two congruent triangles along a diagonal.

 

When calculating with a parallelogram, it is often necessary to use trigonometry and triangle calculations.

 

 

Calculator

Formulas

Side a

$$ a=\frac{Area}{h} $$

$$ a=\frac{Area}{b\cdot sin(A)} $$

$$ a=\frac{P-(2\cdot b)}{2} $$

$$ a = \begin{cases} \sqrt{d_1^2 - h^2} + \sqrt{b^2 - h^2}, & A < 90^\circ \\[6pt] \sqrt{d_1^2 - h^2} - \sqrt{b^2 - h^2}, & A > 90^\circ \end{cases} $$

$$ a = \begin{cases} \sqrt{d_2^2 - h^2} - \sqrt{b^2 - h^2}, & A > 90^\circ \\[6pt] \sqrt{d_2^2 - h^2} + \sqrt{b^2 - h^2}, & A < 90^\circ \end{cases} $$

Side b

$$ b=\frac{h}{sin(A)} $$

$$ b=\frac{Area}{a\cdot sin(A)} $$

$$ b=\frac{P-(2\cdot a)}{2} $$

$$ b = \sqrt{\Bigl(\sqrt{d_1^2 - h^2} - a\Bigr)^2 + h^2} $$

$$ b = \sqrt{\Bigl(a - \sqrt{d_2^2 - h^2}\Bigr)^2 + h^2} $$

Height

$$ h = b \cdot \sin(A) $$

$$ h = \frac{\text{Area}}{a} $$

Diagonal 1

$$ d_1 = \sqrt{a^2 + b^2 - 2ab \cos(A)} $$

Diagonal 2

$$ d_2 = \sqrt{a^2 + b^2 + 2ab \cos(A)} $$

Angle A

$$ \angle A=180-B $$

$$ \angle A=sin^{-1}\biggl(\frac{h}{b}\biggr) $$

$$ \angle A = sin^{-1}\Bigl(\tfrac{\text{Area}}{a \cdot b}\Bigr) $$

$$ \angle A = cos^{-1}\Bigl(\tfrac{a^2 + b^2 - d_1^2}{2ab}\Bigr) $$

$$ \angle A = cos^{-1}\Bigl(\tfrac{d_2^2 - a^2 - b^2}{2ab}\Bigr) $$

Angle B

$$ B = 180 - A $$

$$ B = \sin^{-1}\!\Bigl(\tfrac{h}{a}\Bigr) $$

$$ B = \sin^{-1}\!\Bigl(\tfrac{\text{Area}}{a \cdot b}\Bigr) $$

Area

$$ A = a \cdot h $$

$$ A= a \cdot b \cdot \sin(\angle A) $$

Perimeter

$$ P = 2a + 2b $$

$$ P=2\cdot\biggl(\frac{h}{sin(A)}+\frac{Area}{h}\biggr) $$