SafekipediaA parallelogram is a special four-sided shape. It has two pairs of sides that run parallel to each other. This means the sides stay the same distance apart and never meet.
The opposite sides of a parallelogram are always equal in length. The opposite angles are also equal. This shape is important in Euclidean geometry. Euclidean geometry studies flat shapes and their properties.
Unlike some other four-sided shapes, a parallelogram always has two pairs of parallel sides. If a shape has only one pair of parallel sides, it is called a trapezoid (in American English) or a trapezium (in British English). Parallelograms help us understand many geometric ideas. They are used in many areas of math and science.
The word “parallelogram” comes from ancient Greek. It means “a shape of parallel lines.” In three dimensions, the counterpart to a parallelogram is called a parallelepiped. This shape is like stacking parallelograms together to form a solid.
Special cases
A rectangle is a type of parallelogram with four right angles. A rhombus is a parallelogram where all four sides are the same length. A square is a special kind of rhombus and rectangle, having both equal sides and right angles. There is also a shape called a rhomboid, which has sides of unequal length and angles that are not right angles, but we don't use this term much in math today.
Characterizations
A parallelogram is a four-sided shape where the opposite sides are parallel and the same length. This means that if you measure two sides that face each other, they will be the same length and run in the same direction.
There are several ways to recognize a parallelogram. For example, the opposite angles will also be equal. If you draw a line connecting opposite corners, it will cut the shape into two smaller shapes that are exactly the same. All these properties help us identify and work with parallelograms in geometry.
Other properties
A parallelogram has opposite sides that are parallel and will never meet. The area of a parallelogram is twice the area of a triangle formed by one of its diagonals.
A parallelogram can be turned into another parallelogram through certain changes, and it has rotational symmetry where it looks the same after being turned 180 degrees. If it also has lines of symmetry, it might be a special shape like a rhombus or a square.
Area formula
All area formulas for flat, four-sided shapes also work for parallelograms. For parallelograms, a simple way to find the area uses the base and height. If you know the length of the base (b) and the height (h), the area (K) is just base times height: K = b × h.
There are also other ways to find the area. For example, if you know the lengths of two sides (B and C) and the angle (θ) between them, the area is K = B × C × sin θ. These formulas help us find the space inside any parallelogram.
Proof that diagonals bisect each other
To show that the diagonals of a parallelogram bisect each other, we can look at congruent triangles. Because opposite sides of a parallelogram are equal and parallel, angles created by a line crossing them are also equal. This means that two triangles formed by the diagonals are congruent.
As a result, the parts of the diagonals are equal in length. This proves that the diagonals of a parallelogram bisect each other, meeting at a common midpoint.
Lattice of parallelograms
Parallelograms can fit together to cover a flat surface without any gaps or overlaps. When the sides of the parallelograms are all the same length or the angles are all right angles, the pattern that forms has even more symmetry. These special patterns are known as the four Bravais lattices in 2 dimensions.
| Form | Square | Rectangle | Rhombus | Rhomboid |
|---|---|---|---|---|
| System | Square (tetragonal) | Rectangular (orthorhombic) | Centered rectangular (orthorhombic) | Oblique (monoclinic) |
| Constraints | α=90°, a=b | α=90° | a=b | None |
| Symmetry | p4m, [4,4], order 8n | pmm, [∞,2,∞], order 4n | p1, [∞+,2,∞+], order 2n | |
| Form | ||||
Parallelograms arising from other figures
Varignon parallelogram
Main article: Varignon's theorem
Varignon's theorem says that if you join the middle points of the sides of any four-sided shape, you always make a parallelogram. This parallelogram is called the Varignon parallelogram. If the original shape is simple and does not cross itself, the area of the Varignon parallelogram is exactly half the area of the original shape.
Tangent parallelogram of an ellipse
For an ellipse, two diameters are called conjugate if the tangent line at the end of one diameter is parallel to the other diameter. Each pair of conjugate diameters has a tangent parallelogram, made from the tangent lines at the four ends of these diameters. All such tangent parallelograms for the same ellipse have the same area.
Faces of a parallelepiped
A parallelepiped is a three-dimensional shape where all six faces are parallelograms.
This article is a child-friendly adaptation of the Wikipedia article on Parallelogram, available under CC BY-SA 4.0.
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