Rectangle: Detailed Explanation
A rectangle is a two-dimensional quadrilateral that belongs to the parallelogram family. It has a variety of important geometric properties that make it distinct from other shapes.
Properties of a Rectangle:
- Opposite sides are equal: A rectangle has two pairs of opposite sides, and each pair of opposite sides is equal in length.
- All angles are 90 degrees: Each of the four angles in a rectangle is a right angle (90°).
- Opposite sides are parallel: The two opposite sides of a rectangle are parallel, just like in a parallelogram.
- Diagonals:
- The diagonals of a rectangle are equal in length.
- The diagonals bisect each other, but they do not necessarily bisect the angles.
- The diagonals do not intersect at right angles (unlike a square or rhombus).
- Area and Perimeter:
- Area = Length × Width (where the length and width are the two adjacent sides of the rectangle).
- Perimeter = 2 × (Length + Width).
- Symmetry:
- A rectangle has two lines of symmetry: one that divides it horizontally and one that divides it vertically.
- A rectangle also has rotational symmetry at 180°.
Comparison of Rectangle with Other Figures:
| Figure | Similarities with Rectangle | Differences from Rectangle |
|---|---|---|
| Square | – Opposite sides are parallel and equal. – All angles are 90°. | – In a square, all four sides are equal, while in a rectangle only opposite sides are equal. – The diagonals of a square bisect each other at right angles. |
| Rhombus | – Opposite sides are parallel. – Both are quadrilaterals. | – A rhombus has all sides equal, but its angles are not 90°. – The diagonals of a rhombus bisect each other at right angles, but they are not equal. |
| Parallelogram | – Opposite sides are parallel and equal. | – The angles in a parallelogram are not necessarily 90°. – The diagonals are not necessarily equal in length and do not bisect at right angles. |
| Trapezium (Trapezoid) | – Both are quadrilaterals. | – In a trapezium, only one pair of sides is parallel, and the other pair is not. – Sides and angles are not equal in a trapezium. |
Rectangle vs Square:
- Commonality: Both a rectangle and a square have four right angles, and opposite sides are parallel.
- Difference: A rectangle has opposite sides that are equal, while in a square, all sides are equal. The diagonals of a square intersect at right angles, while in a rectangle, they do not.
Rectangle vs Rhombus:
- Commonality: Both have opposite sides that are parallel.
- Difference: In a rhombus, all sides are equal, but the angles are not necessarily 90 degrees, unlike in a rectangle. The diagonals of a rhombus bisect at right angles, but in a rectangle, the diagonals do not bisect at right angles.
Rectangle vs Parallelogram:
- Commonality: Both are quadrilaterals, and in both, opposite sides are parallel and equal.
- Difference: The key difference is that a rectangle has four right angles, whereas in a parallelogram, the angles are not necessarily 90°. Also, the diagonals of a parallelogram are not necessarily equal, while in a rectangle, they are.
Rectangle vs Trapezium:
- Commonality: Both are quadrilaterals (four-sided figures).
- Difference: In a rectangle, opposite sides are parallel and equal, while in a trapezium, only one pair of sides is parallel. The angles and sides are also not necessarily equal in a trapezium.
Real-Life Examples of a Rectangle:
- Books: Most book covers are shaped like rectangles.
- Doors: Many doors have a rectangular shape.
- Mobile Phones: The screens of most smartphones are designed in a rectangular shape.
Important Formulas:
- Area = Length × Width.
- Perimeter = 2 × (Length + Width).
- Diagonal = (Length2+Width2)\sqrt{(Length^2 + Width^2)}.