Quadrilateral formulas area perimeter and solved examples
The concept of quadrilaterals plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you are preparing for board exams or competitive test papers, understanding quadrilaterals, their types, and properties gives you an edge in geometry questions.
What Is a Quadrilateral?
A quadrilateral is a two-dimensional polygon with four sides, four angles, and four vertices. The word itself comes from "quadra" (meaning four) and "latus" (meaning side). You’ll find this concept applied in areas such as mensuration, symmetry, coordinate geometry, and tiling patterns in daily life. Every closed figure made by joining four non-collinear points is a quadrilateral. The sum of its interior angles is always 360°.
Properties of Quadrilaterals
- It has exactly four sides and four vertices.
- The sum of all interior angles is always 360°.
- There are two diagonals in every quadrilateral.
- Some or all sides may be equal (depending on the type).
- Some or all angles may be equal or right angles (90°) in special cases.
- Diagonals may be equal, bisect each other, or be perpendicular in certain types.
- Quadrilaterals can be classified according to parallel sides, angle measures, and diagonal properties.
Types of Quadrilaterals
There are several types of quadrilaterals, each with unique qualities. The most common types are shown in the table below:
| Type | Description | Key Properties |
|---|---|---|
| Square | All sides equal; all angles 90° | Diagonals equal, bisect at right angles; all sides parallel in pairs |
| Rectangle | Opposite sides equal; all angles 90° | Diagonals equal and bisect each other |
| Parallelogram | Opposite sides parallel and equal | Diagonals bisect each other; opposite angles equal |
| Rhombus | All sides equal; opposite sides parallel | Diagonals bisect at right angles but are not equal |
| Trapezium (Trapezoid) | One pair of sides parallel | Sum of angles on same side of parallel lines is 180° |
| Kite | Two pairs of adjacent sides equal | One diagonal bisects the other; diagonals are perpendicular |
| Irregular Quadrilateral | No equal sides or angles | No specific symmetry; only angle sum property holds |
Note: For irregular quadrilaterals, divide into two triangles and use Heron's formula or coordinate geometry.
Step-by-Step Illustration: Example Problem
Find the area of a trapezium with parallel sides 10 cm and 6 cm and height 5 cm.
1. Write the formula:Area = (1/2) × (sum of parallel sides) × height
2. Substitute values:
Area = (1/2) × (10 + 6) × 5
3. Calculate sum of parallel sides: 10 + 6 = 16
4. Multiply: (1/2) × 16 = 8
5. Area = 8 × 5 = 40
6. Final Answer: 40 cm²
Speed Trick or Exam Tip
To quickly identify the type of quadrilateral in MCQs, look first for parallel sides and right angles. If all sides are equal and all angles are 90°, it’s a square. If only opposite sides are equal and all angles are 90°, it’s a rectangle. Use shape properties as clues!
Relation to Other Concepts
The idea of quadrilaterals connects closely with triangles, polygons, and coordinate geometry. Mastering these shapes helps in finding areas, perimeters, and solving advanced geometry like parallelograms and cyclic quadrilaterals.
Try These Yourself
- Draw and label all types of quadrilaterals.
- Find the area of a rhombus with diagonals 8 cm and 6 cm.
- Which quadrilateral has exactly one pair of parallel sides?
- If one angle in a rectangle is 90°, what about the others?
Frequent Errors and Misunderstandings
- Confusing parallelogram with rectangle or rhombus
- Forgetting all angles in a quadrilateral sum to 360°
- Assuming all diagonals are equal for every quadrilateral
- Applying the wrong area formula to irregular shapes
Classroom Tip
A quick way to remember the sum of angle property: divide a quadrilateral into two triangles; 2 × 180° = 360°. Vedantu’s teachers often draw this to help students in live and online classes.
We explored quadrilaterals—from the basic definition, properties, and types, to area formulas and common tricks. Continue practicing with Vedantu for exam-ready confidence and clear understanding of geometry concepts.
Further Learning & Internal Links
- Types of Quadrilaterals—Deep dive into all quadrilateral types with diagrams and characteristics for each.
- Area of a Quadrilateral—Explore area formulas for squares, rectangles, rhombus, trapezium, and more.
- Quadrilaterals MCQs—Test yourself with practice quizzes based on recent exams.
- Quadrilateral Angle Sum Property—Proof, diagrams, and practice problems about the angle sum of 360°.
FAQs on Quadrilateral Concepts Definition Types and Properties
1. What is a quadrilateral?
A quadrilateral is a polygon with four sides, four vertices, and four angles. It is a closed 2D shape formed by joining four line segments end to end. Examples of quadrilaterals include:
- Square
- Rectangle
- Parallelogram
- Rhombus
- Trapezium (Trapezoid)
2. What is the sum of the interior angles of a quadrilateral?
The sum of the interior angles of any quadrilateral is 360°. This can be understood by dividing a quadrilateral into two triangles. Since each triangle has an angle sum of 180°:
- 2 × 180° = 360°
- 360° − (80° + 90° + 100°) = 90°
3. What is the formula for the area of a quadrilateral?
The area of a quadrilateral depends on its type because there is no single formula for all quadrilaterals. Common area formulas include:
- Square: Area = side²
- Rectangle: Area = length × width
- Parallelogram: Area = base × height
- Trapezium: Area = ½ × (sum of parallel sides) × height
4. What are the different types of quadrilaterals?
The main types of quadrilaterals are classified based on their sides and angles. These include:
- Square: All sides equal and all angles 90°
- Rectangle: Opposite sides equal and all angles 90°
- Parallelogram: Opposite sides parallel and equal
- Rhombus: All sides equal, opposite angles equal
- Trapezium (Trapezoid): At least one pair of parallel sides
- Kite: Two pairs of adjacent equal sides
5. What is the difference between a square and a rectangle?
The key difference is that a square has all four sides equal, while a rectangle has only opposite sides equal. Both shapes have four right angles (90°).
- Square: All sides equal, diagonals equal and perpendicular
- Rectangle: Opposite sides equal, diagonals equal but not perpendicular
6. How do you find the perimeter of a quadrilateral?
The perimeter of a quadrilateral is found by adding the lengths of all four sides. The formula is:
- Perimeter = a + b + c + d
- Perimeter = 5 + 7 + 6 + 8 = 26 cm
7. What are the properties of a parallelogram?
A parallelogram is a quadrilateral in which opposite sides are parallel and equal. Its main properties are:
- Opposite sides are equal and parallel
- Opposite angles are equal
- Consecutive angles add up to 180°
- Diagonals bisect each other
8. What is a cyclic quadrilateral?
A cyclic quadrilateral is a quadrilateral whose four vertices lie on a single circle. The key property is:
- Opposite angles are supplementary, meaning they add up to 180°
9. How do you find a missing angle in a quadrilateral?
You find a missing angle in a quadrilateral by using the fact that the interior angles add up to 360°. Follow these steps:
- Add the given angles
- Subtract their sum from 360°
- Total of known angles = 280°
- Missing angle = 360° − 280° = 80°
10. What is the difference between a trapezium and a parallelogram?
The main difference is that a trapezium has at least one pair of parallel sides, while a parallelogram has two pairs of parallel sides. Key distinctions include:
- Trapezium: Only one pair of opposite sides is parallel
- Parallelogram: Both pairs of opposite sides are parallel and equal
- In a parallelogram, opposite angles are equal; this is not necessarily true for a trapezium
