Quadrilaterals – Four Sides, Many Shapes
A quadrilateral is any closed shape with four straight sides and four angles. Quadrilaterals are everywhere: the rooms we live in, the screens we look at, and the tiles on the floor are all quadrilaterals.
Key Properties of All Quadrilaterals
Every quadrilateral has four sides, four vertices, and two diagonals. The interior angles of any quadrilateral always sum to 360°.
Types of Quadrilaterals
| Shape | Sides | Angles | Diagonals |
|---|---|---|---|
| Square | All 4 equal; all parallel in pairs | All 90° | Equal, bisect at 90° |
| Rectangle | Opposite sides equal; all parallel in pairs | All 90° | Equal, bisect each other |
| Rhombus | All 4 equal; opposite sides parallel | Opposite angles equal | Bisect at 90°, not equal |
| Parallelogram | Opposite sides equal and parallel | Opposite angles equal | Bisect each other |
| Trapezium | One pair of parallel sides | Co-interior angles sum to 180° | Not equal, do not bisect |
| Kite | Two pairs of adjacent equal sides | One pair of equal opposite angles | One bisects the other at 90° |
Angle Sum Property
A quadrilateral has three angles of 85°, 100°, and 95°. Find the fourth.
Fourth angle = 360 − 85 − 100 − 95 = 80°.
A parallelogram has one angle of 65°. Find all four angles.
Opposite angles equal: two angles are 65°. Adjacent angles are supplementary: 180 − 65 = 115°. Angles: 65°, 115°, 65°, 115°.
Areas of Quadrilaterals
| Shape | Area Formula |
|---|---|
| Square | side² |
| Rectangle | length × width |
| Parallelogram | base × perpendicular height |
| Rhombus | ½ × d₁ × d₂ (product of diagonals) |
| Trapezium | ½ × (a + b) × h |
| Kite | ½ × d₁ × d₂ |
Key Takeaways
- All quadrilaterals have four sides and interior angles summing to 360°.
- A square is the most special: all sides equal AND all angles 90°.
- A rectangle has all 90° angles but sides need not all be equal.
- A rhombus has all equal sides but angles need not be 90°.
- A trapezium has exactly one pair of parallel sides.
Practice Questions
- Name four types of quadrilateral and list one key property of each.
- A quadrilateral has angles 3x, 2x, x, and 2x. Find each angle and identify the shape if possible.
- Calculate the area of a trapezium with parallel sides 6 cm and 10 cm and height 4 cm.
- A rhombus has diagonals of 8 cm and 6 cm. Find its area.
- Is every square a rectangle? Is every rectangle a square? Explain.