Quadrilaterals

A quadrilateral is a polygon with four sides, four vertices, and four angles. The sum of the interior angles of a quadrilateral is always 360°.

Types of Quadrilaterals

• Parallelogram: Opposite sides are parallel and equal, opposite angles are equal.
• Rectangle: All angles are 90°, opposite sides are equal and parallel.
• Square: All sides are equal, all angles are 90°, opposite sides are parallel.
• Rhombus: All sides are equal, opposite angles are equal, diagonals bisect at right angles.
• Trapezium (Trapezoid): Only one pair of opposite sides is parallel.
• Kite: Two pairs of adjacent sides are equal, one pair of opposite angles is equal.

Properties of Quadrilaterals

• Sum of all interior angles = 360°
• Diagonals may or may not be equal, depending on the type
• Some quadrilaterals have lines of symmetry (e.g., square, rectangle, rhombus)

Important Formulas

• Area of Rectangle: length × breadth
• Area of Square: side × side
• Area of Parallelogram: base × height
• Area of Rhombus: (1/2) × (product of diagonals)
• Area of Trapezium: (1/2) × (sum of parallel sides) × height

Diagonals in Quadrilaterals

• Parallelogram: Diagonals bisect each other.
• Rectangle: Diagonals are equal and bisect each other.
• Square: Diagonals are equal, bisect each other at 90°.
• Rhombus: Diagonals bisect each other at 90°, but are not equal.
• Kite: One diagonal bisects the other at 90°.

Sample Problems

1. Find the area of a rectangle with length 8 cm and breadth 5 cm.
   Solution: Area = 8 × 5 = 40 cm²
2. A square has a side of 6 cm. What is its area?
   Solution: Area = 6 × 6 = 36 cm²
3. Find the area of a parallelogram with base 10 cm and height 4 cm.
   Solution: Area = 10 × 4 = 40 cm²
4. The diagonals of a rhombus are 12 cm and 9 cm. Find its area.
   Solution: Area = (1/2) × 12 × 9 = 54 cm²
5. Find the area of a trapezium with parallel sides 7 cm and 13 cm, and height 5 cm.
   Solution: Area = (1/2) × (7 + 13) × 5 = (1/2) × 20 × 5 = 50 cm²

Practice Questions

1. List all types of quadrilaterals and write one property of each.
2. Draw a parallelogram and mark its diagonals. What do you observe?
3. If the area of a square is 81 cm², what is the length of its side?
4. Find the area of a rhombus whose diagonals are 10 cm and 8 cm.
5. Which quadrilaterals have all sides equal?

Summary

• Quadrilaterals are four-sided polygons with a total angle sum of 360°.
• Different types include parallelogram, rectangle, square, rhombus, trapezium, and kite.
• Each type has unique properties related to sides, angles, and diagonals.
• Area formulas vary by type and are important for solving real-life problems.