What Is a Parallelogram?
A parallelogram is a four-sided polygon (quadrilateral) with two pairs of parallel sides. Opposite sides are equal in length, and opposite angles are equal. The sum of adjacent angles is 180 degrees. Rectangles, rhombuses, and squares are all special types of parallelograms.
Parallelogram Area Formulas
Base and Height
The most straightforward method using the base and perpendicular height.
Two Sides and Angle
Using two adjacent sides and the included angle between them.
Diagonals and Angle
Using the diagonals and the angle between them.
Other Parallelogram Properties
Perimeter
The perimeter is the sum of all sides: P = 2(a + b), where a and b are the lengths of two adjacent sides.
Diagonals
The diagonals of a parallelogram bisect each other but are generally not equal. Their lengths can be found using:
- d1 = sqrt(a² + b² + 2ab cos(theta))
- d2 = sqrt(a² + b² - 2ab cos(theta))
The parallelogram law states: d1² + d2² = 2(a² + b²).
Special Cases
- Rectangle: A parallelogram where all angles are 90 degrees (theta = 90 degrees, sin(90) = 1).
- Rhombus: A parallelogram where all sides are equal (a = b).
- Square: A parallelogram that is both a rectangle and a rhombus (a = b, theta = 90 degrees).
Real-World Applications
Parallelograms appear in architecture (tilted window frames), engineering (force diagrams using the parallelogram law of vector addition), art (perspective drawing), and everyday objects like erasers and certain table designs. Understanding their properties is essential for structural analysis and design.