A parallelogram is a quadrilateral (a four-sided polygon) with two pairs of parallel sides. In a parallelogram, opposite sides are equal in length, and opposite angles are equal in measure. The name comes from the Greek 'parallelogrammon', which means 'bounded by parallel lines'. This shape is fundamental in Euclidean geometry and serves as a basis for other quadrilaterals like rectangles, rhombuses, and squares.
A parallelogram with vertices labeled A, B, C, and D in counterclockwise order. The base is side AB, with length 'b'. The adjacent side is AD, with length 'a'. The perpendicular height, 'h', is shown as a dashed line from vertex D down to the base AB, forming a right angle. The interior angle at vertex A is labeled α (alpha). The two diagonals, AC and BD, intersect inside the shape.
Side Properties: Opposite sides are parallel and equal in length.
Angle Properties: Opposite angles are equal. Consecutive (adjacent) angles are supplementary, meaning they add up to 180°.
Diagonal Properties: The diagonals bisect each other, meaning they cross at their midpoint. The diagonals are not generally equal in length.
Symmetry: A parallelogram has point symmetry about the intersection of its diagonals (rotational symmetry of order 2). It does not typically have line symmetry unless it is a rhombus or a rectangle.
The area formula for a parallelogram, A = base × height, can be proven by dissecting and rearranging the shape into a rectangle.
1. Start with a parallelogram with base 'b' and height 'h'.
2. Drop a perpendicular line from one of the top vertices to the base. This cuts off a right-angled triangle from one side of the parallelogram.
3. Translate (slide) this triangle to the other side of the parallelogram. It will fit perfectly because the opposite sides are parallel and equal in length.
4. The new shape formed is a rectangle with a length equal to the parallelogram's base 'b' and a width equal to the parallelogram's height 'h'.
Since the area of the shape has not changed during this transformation (the principle of Cavalieri), the area of the original parallelogram is also equal to b × h.
Engineering & Mechanics: Parallelograms are used to represent the addition of forces and velocities. The resultant vector of two forces acting at a point is the diagonal of the parallelogram formed by the two force vectors. This is known as the Parallelogram of Forces.
Computer Graphics: In 2D and 3D graphics, shear transformations and other affine mappings transform squares and rectangles into parallelograms. This is used for creating perspective, slanting objects, and texture mapping.
Architecture & Construction: The structural integrity of many frameworks, such as trusses and bridges, relies on the rigid properties of triangles, which are often arranged in patterns that form parallelograms. Some modern architectural designs use parallelogram shapes for windows, facades, and floor plans.
Physics & Crystallography: In solid-state physics, the unit cell of a crystal lattice in two dimensions is often a parallelogram, defining the repeating structure of atoms within the material.
Architecture: The sloping sides of the John Hancock Center in Chicago form large parallelograms, a distinctive feature of its structural X-bracing design.
Mechanical Linkages: The mechanism that allows a bus door to open outwards while remaining parallel to the side of the bus, or the design of some car jacks and retractable gates, often uses a system of linked bars that form moving parallelograms.
Art and Design: Artists like M.C. Escher used tessellations of parallelograms to create intricate patterns. In graphic design, the shape is used to convey a sense of motion, speed, or dynamism.
A parallelogram is a convex quadrilateral with parallel opposite sides. There are several special types of parallelograms, classified by their angle and side properties:
| Type | Properties |
|---|---|
| Rectangle | All four angles are right angles (90°). |
| Rhombus | All four sides are equal in length. |
| Square | All four angles are right angles AND all four sides are equal. (A square is both a rectangle and a rhombus). |
Using the side length instead of the height for the area calculation. The area is base times perpendicular height (A = b × h), not base times the length of the slanted side.
Assuming the diagonals are equal. The diagonals of a parallelogram are only equal if it is a rectangle. For a general parallelogram, one diagonal is longer than the other.
Assuming the diagonals bisect the angles. The diagonals only bisect the vertex angles if the parallelogram is a rhombus.