Parallelogrammien

Parallelogrammien is the genitive plural form of the Finnish noun parallelogrammi, the geometric figure known in English as a parallelogram. In mathematical writing, parallelogrammien appears in phrases such as parallelogrammien sivut or parallelogrammien ominaisuudet, indicating possession or relation to several parallelograms. The article outlines the basic geometry of parallelograms.

Definition and basic properties: A parallelogram is a simple quadrilateral with two pairs of opposite sides that are parallel. Opposite sides are equal in length; opposite angles are equal; consecutive angles are supplementary. The diagonals intersect at their midpoints, so they bisect each other.

Special cases: If all angles are right angles, the parallelogram is a rectangle. If all sides are equal, it is a rhombus. If it has both properties, it is a square.

Area, perimeter, and diagonals: The area is base × height, or a b sin θ when adjacent sides have lengths a and b and include angle θ. The perimeter is 2(a + b). The diagonals d1 and d2 have lengths given by d1^2 = a^2 + b^2 + 2ab cos θ and d2^2 = a^2 + b^2 − 2ab cos θ, and they bisect each other.

Applications and notes: Parallelograms appear in geometry problems, vector representations, and design. The term parallelogrammien is linguistic rather than a distinct mathematical term; the mathematical concept is parallelogram (parallelogrammi) in Finnish.