Trapezium: Detailed Explanation
A trapezium (called a trapezoid in American English) is a quadrilateral, meaning it has four sides. What makes a trapezium unique among quadrilaterals is that it has only one pair of opposite sides that are parallel. The other pair of sides is non-parallel.
Properties of a Trapezium:
- One pair of opposite sides is parallel: In a trapezium, only one pair of sides is parallel. These sides are called the bases of the trapezium.
- Non-parallel sides: The other two sides, which are not parallel, are called the legs of the trapezium.
- Angles: The sum of the interior angles of any quadrilateral is always 360°. In a trapezium, the adjacent angles along each leg are supplementary (sum to 180°). The angles between the base and legs are not necessarily equal.
- Types of Trapezium:
- Isosceles Trapezium: Both non-parallel sides (legs) are equal in length, and the base angles (angles along the parallel sides) are equal.
- Scalene Trapezium: A trapezium where all sides and angles are unequal.
- Right-Angled Trapezium: At least one of the angles is a right angle (90°).
- Diagonals: In an isosceles trapezium, the diagonals are equal in length, but in a general trapezium, the diagonals are unequal.
- Area and Perimeter:
- Area = 12×(Base1+Base2)×Height\frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}, where Base1 and Base2 are the lengths of the two parallel sides, and the height is the perpendicular distance between them.
- Perimeter = Sum of all four sides.
Comparison of Trapezium with Other Figures:
| Figure | Similarities with Trapezium | Differences from Trapezium |
|---|---|---|
| Square | – Both are quadrilaterals. | – A square has all sides equal and all angles 90°. – In a trapezium, only one pair of sides is parallel, while in a square, opposite sides are parallel. |
| Rectangle | – Both are quadrilaterals. – Both have parallel sides. | – In a rectangle, both pairs of opposite sides are parallel, whereas in a trapezium, only one pair of opposite sides is parallel. – A rectangle has all angles equal to 90°, unlike a trapezium. |
| Rhombus | – Both are quadrilaterals. | – In a rhombus, all sides are equal, and opposite sides are parallel. – A rhombus has diagonals that bisect at 90°, while in a trapezium, diagonals are not equal or perpendicular. |
| Parallelogram | – Both are quadrilaterals. | – In a parallelogram, both pairs of opposite sides are parallel, while in a trapezium, only one pair of sides is parallel. – Diagonals bisect each other in a parallelogram but not in a trapezium. |
| Kite | – Both are quadrilaterals. | – A kite has two pairs of adjacent equal sides, while a trapezium has only one pair of parallel sides. – The diagonals of a kite bisect at 90°, whereas in a trapezium, diagonals do not bisect at right angles. |
Trapezium vs Square:
- Commonality: Both are four-sided shapes (quadrilaterals).
- Difference: A square has all sides equal and all angles at 90°, while a trapezium has only one pair of parallel sides, and the other pair of sides are unequal and non-parallel.
Trapezium vs Rectangle:
- Commonality: Both have parallel sides.
- Difference: A rectangle has two pairs of opposite sides that are parallel and equal in length, and all four angles are 90°. In a trapezium, only one pair of opposite sides is parallel, and the angles and side lengths vary.
Trapezium vs Rhombus:
- Commonality: Both are quadrilaterals.
- Difference: A rhombus has all sides equal in length, and opposite sides are parallel. In contrast, a trapezium has only one pair of parallel sides, and the other sides are unequal. Also, in a rhombus, the diagonals intersect at right angles, which is not the case in a trapezium.
Trapezium vs Parallelogram:
- Commonality: Both are quadrilaterals.
- Difference: A parallelogram has two pairs of opposite sides that are parallel, while a trapezium has only one pair of parallel sides. The diagonals in a parallelogram bisect each other, while in a trapezium, they do not.
Trapezium vs Kite:
- Commonality: Both are quadrilaterals.
- Difference: A kite has two pairs of adjacent equal sides, while a trapezium has only one pair of parallel sides. The diagonals of a kite intersect at right angles, while in a trapezium, the diagonals may be unequal and do not bisect at right angles.
Real-Life Examples of a Trapezium:
- Bridges: Certain bridges have trapezium-shaped structures, particularly in their side supports.
- Trapezoidal Prism: Some architectural designs use trapezium shapes in modern buildings.
- Table Tops: Certain tables, particularly those designed for small or angled spaces, may have a trapezium shape.
Important Formulas:
- Area = 12×(Base1+Base2)×Height\frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}, where Base1 and Base2 are the lengths of the two parallel sides, and the height is the perpendicular distance between them.
- Perimeter = Sum of all four sides.