Parallelogram: Definition, Properties, and Calculations

A parallelogram is a quadrilateral whose opposite sides are parallel in pairs, meaning they lie on parallel lines.

Special Cases of Parallelograms

Special cases of parallelograms include:

• Rectangle: A parallelogram with four right angles.

• Square: A parallelogram with four right angles and all sides equal.

• Rhombus: A parallelogram with all sides equal.

What Does a Parallelogram Look Like?

A parallelogram with a height drawn to one of the bases and diagonals.

In the figure above, the parallelogram is marked with blue lines.

Elements of a Parallelogram

• ABCD: A parallelogram whose opposite sides are parallel in pairs (AB is parallel to CD, and BC is parallel to AD).

• BH: The height of the parallelogram dropped from point B to the base AD (marked in red in the figure).

• AC and BD: The diagonals of the parallelogram.

Properties of a Parallelogram

• Opposite sides of a parallelogram are equal.

• Opposite angles of a parallelogram are equal.

• The diagonals of a parallelogram intersect and are bisected by the point of intersection. The point of intersection
  of the diagonals is called the center of symmetry of the parallelogram.

• The diagonal divides the parallelogram into two equal triangles.

• The sum of the angles adjacent to one side is 180°.

• The sum of all angles is 360°.

• The midlines of a parallelogram intersect at the point of intersection of its diagonals and are divided in half by this point.

• The sum of the squares of the diagonals of a parallelogram is equal to twice the sum of the squares of its sides.

Parallelogram Characteristics

A quadrilateral ABCD is a parallelogram if one of the following conditions is met:

• Opposite sides are pairwise equal.
• Opposite sides are pairwise parallel and equal.
• Opposite angles are pairwise equal.
• Diagonals are bisected at their intersection point.
• The sum of adjacent angles is 180 degrees.
• Two sides are equal and parallel.

How to Find the Area of a Parallelogram

• The area of a parallelogram is equal to the product of the length of one of its sides by the height dropped to this side (1).

• The area of a parallelogram is equal to the product of its two adjacent sides by the sine of the angle between them (2,3).

• The area of a parallelogram is equal to half the product of its diagonals by the sine of the angle between them (4).

• The area of a parallelogram can also be found using Heron's formula, considering one of the diagonals as a triangle
  and calculating the doubled area of this triangle (5).

How to Find the Sides of a Parallelogram

The sides of a parallelogram can be found through:

1. The sizes of the diagonals and the angle between them (1,2).

2. The lengths of the diagonals and one of the sides (3,4).

3. The height dropped onto the side and the angle between the sides (5,6).

4. The area and height dropped onto a given side (7,8).

How to Find the Diagonals of a Parallelogram

1. The lengths of its sides and the cosine of the angle between them (1,2,3,4).

2. The lengths of the sides and the size of the second diagonal (5,6).

3. The area, the length of the second diagonal, and the angle between them (7,8).

How to Find the Perimeter of a Parallelogram

The perimeter of a parallelogram can be found:

1. By its sides (1).

2. By one of the sides and the length of two diagonals (2,3).

3. By the side, height, and angle between the sides (4,5,6).