Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides. Opposite sides and opposite angles are equal.

h a b α

Special Cases

A rectangle is a parallelogram with all angles 90°. A rhombus has all sides equal. A square is both a rectangle and a rhombus.

Basic Formulas

S = a · h Area (base × height)
S = a · b · sin(α) Area using two sides and the included angle
P = 2(a + b) Perimeter (twice the sum of adjacent sides)
d₁² + d₂² = 2(a² + b²) Parallelogram law (relation between sides and diagonals)

Example

Given: base a = 10 cm, side b = 6 cm, height h = 5 cm, angle α = 30°

Area (base×height): S = 10 · 5 = 50 cm²

Area (sine formula): S = 10 · 6 · sin(30°) = 10 · 6 · 0.5 = 30 cm²

Note: The sine formula gives the area using slant height and angle; the base×height formula uses the perpendicular height. These correspond to different orientations.

Perimeter: P = 2(10 + 6) = 2 · 16 = 32 cm

Notes

The parallelogram law states that the sum of squares of the diagonals equals twice the sum of squares of the sides. This is a generalization of the Pythagorean theorem for rectangles.

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