Parallelogram Trapezium and Kite Definition Formulas Properties and Solved Examples
The concept of Parallelogram, Trapezium, and Kite plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. These are all special types of quadrilaterals, each with its unique properties about sides, angles, and diagonals. Mastering their differences is essential for scoring well and solving geometry problems quickly in school and competitive exams.
What Is Parallelogram, Trapezium, and Kite?
A parallelogram is a quadrilateral where both pairs of opposite sides are parallel and equal in length. The trapezium (also known as a trapezoid in some countries) is a four-sided shape with only one pair of parallel sides. A kite is a quadrilateral with two pairs of adjacent sides equal in length. You’ll find these concepts applied in geometry, mensuration, and logical reasoning problems.
Properties Comparison Table
| Property | Parallelogram | Trapezium | Kite |
|---|---|---|---|
| Definition | Opposite sides parallel and equal | One pair of sides parallel | Two pairs of adjacent sides equal |
| Equal Sides | Opposite sides | Only in isosceles trapezium | Adjacent sides (two pairs) |
| Equal Angles | Opposite angles | Base angles (isosceles only) | One pair of opposite angles |
| Diagonals | Bisect each other | May not bisect | One diagonal bisects the other at 90° |
| Angle Sum | 360° | 360° | 360° |
| Parallel Sides | 2 pairs | 1 pair | None |
Key Formulas for Parallelogram, Trapezium, and Kite
Parallelogram
Perimeter = 2 × (sum of adjacent sides)
Trapezium
Perimeter = sum of all four sides
Kite
Perimeter = 2 × (sum of distinct side lengths)
How to Identify: Parallelogram, Trapezium, or Kite?
Use these tips during exams or quick revision time:
- If both pairs of sides are parallel, it is a parallelogram.
- If only one pair of opposite sides is parallel, it’s a trapezium.
- If two pairs of sides are equal but the equal ones are adjacent, you have a kite.
Angle Properties and Relationships
Sum of internal angles in any quadrilateral (including parallelogram, trapezium, and kite) is always 360°.
- In a parallelogram, opposite angles are equal, consecutive angles are supplementary (add up to 180°).
- In a kite, one pair of opposite angles are equal (between unequal sides).
- In a trapezium, the sum of angles between each leg and the parallel sides is 180°.
Step-by-Step Illustration: Solved Examples
1. Area of a Parallelogram
Given base = 8 cm, height = 5 cm.
Find the area.
2. Substitute values: 8 × 5 = 40
3. Final Answer: 40 cm²
2. Area of a Trapezium
Given parallel sides: 10 cm and 6 cm, height = 4 cm.
Find the area.
2. Substitute values: ½ × (10 + 6) × 4 = ½ × 16 × 4 = 8 × 4 = 32
3. Final Answer: 32 cm²
3. Perimeter of a Kite
Sides are 7 cm and 9 cm.
Find the perimeter.
2. Substitute values: 2 × (7 + 9) = 2 × 16 = 32
3. Final Answer: 32 cm
Speed Trick or Vedic Shortcut
For parallelogram, trapezium, and kite area questions, quickly check if you know the formula before starting. Repeat the trick: Area of kite is always half the product of diagonals. For trapezium, average the parallel sides and multiply by height. These small techniques can help save time in exams. Vedantu often shares such tips in maths trick lessons and live classes.
Try These Yourself
- Find the area of a parallelogram with base 12 cm and height 7 cm.
- If a kite’s diagonals are 8 cm and 10 cm, what is its area?
- Which of the following is not a quadrilateral: parallelogram, kite, triangle, trapezium?
- Among the given quadrilaterals, which has only one pair of parallel sides?
Frequent Errors and Misunderstandings
- Confusing parallelogram with the kite and trapezium, especially with regards to parallel sides.
- Applying the area formula for kite or parallelogram incorrectly (mixing up diagonals and base/height).
- Assuming all kites have equal diagonals (they only cross at ninety degrees, not always equal length).
Relation to Other Concepts
Understanding parallelogram, trapezium, and kite helps with recognizing all types of quadrilaterals, calculating more complex shapes, and extends to topics like areas of triangles and parallelograms or figures with symmetry, which are key in advanced geometry, physics, and design.
Classroom Tip
A quick way to remember: “A parallelogram has both pairs of opposite sides parallel; a trapezium has only one; a kite has adjacent sides equal.” Vedantu’s teachers use “P-T-K” (Parallel-Two-One, Kite-Adjacent) as a mnemonic device for easy recall during class.
Summary and Exam Tips
- Always check the parallel sides to distinguish between parallelogram and trapezium.
- Don’t mix up base and diagonal formulas when working out areas.
- Use property tables for fast revision before exams.
- For MCQs, remember the unique diagonal rules: parallelogram (bisect each other), kite (one bisects at right angle).
We explored Parallelogram, Trapezium, and Kite—from definitions, key properties, formulas, solved examples, and frequent mistakes. Keep practicing with Vedantu and check out in-depth guides like Properties of Parallelogram to strengthen your understanding and excel in your maths exams!
FAQs on Parallelogram Trapezium and Kite Concepts and Key Properties
1. What is a parallelogram in geometry?
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. It has the following key properties:
- Opposite sides are equal and parallel.
- Opposite angles are equal.
- Consecutive angles are supplementary (sum to 180°).
- Diagonals bisect each other.
2. What is the formula for the area of a parallelogram?
The area of a parallelogram is given by the formula Area = base × height. Here:
- Base (b) is any side of the parallelogram.
- Height (h) is the perpendicular distance between the base and the opposite side.
3. What is a trapezium in geometry?
A trapezium is a quadrilateral with exactly one pair of opposite sides parallel. The parallel sides are called bases, and the non-parallel sides are called legs. A special type is the isosceles trapezium, where the non-parallel sides are equal and base angles are equal. Trapeziums are commonly used in area and coordinate geometry problems.
4. What is the formula for the area of a trapezium?
The area of a trapezium is given by Area = ½ × (sum of parallel sides) × height. In formula form:
- Area = ½ × (a + b) × h
5. What is a kite in geometry?
A kite is a quadrilateral with two pairs of adjacent sides equal in length. Its important properties include:
- Two pairs of adjacent equal sides.
- One pair of opposite angles are equal.
- Diagonals intersect at 90°.
- One diagonal bisects the other.
6. What is the formula for the area of a kite?
The area of a kite is given by Area = ½ × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals. Since the diagonals of a kite are perpendicular, this formula works directly. For example, if d₁ = 8 cm and d₂ = 6 cm, then Area = ½ × 8 × 6 = 24 cm².
7. What is the difference between a parallelogram and a trapezium?
The main difference between a parallelogram and a trapezium is the number of parallel sides.
- A parallelogram has two pairs of parallel sides.
- A trapezium has only one pair of parallel sides.
- In a parallelogram, opposite sides are equal.
- In a trapezium, non-parallel sides are generally unequal (except in isosceles trapezium).
8. Are the diagonals of a parallelogram equal?
The diagonals of a parallelogram are not necessarily equal, but they always bisect each other. In a general parallelogram, the diagonals are of different lengths. However, in special cases:
- In a rectangle, diagonals are equal.
- In a square, diagonals are equal and perpendicular.
9. How do you prove that a quadrilateral is a parallelogram?
A quadrilateral is a parallelogram if any one of the following conditions is satisfied:
- Both pairs of opposite sides are parallel.
- Both pairs of opposite sides are equal.
- Diagonals bisect each other.
- One pair of opposite sides is both equal and parallel.
10. What are the key properties of parallelogram, trapezium, and kite?
The key properties of parallelogram, trapezium, and kite are based on their sides, angles, and diagonals.
- Parallelogram: Opposite sides parallel and equal; diagonals bisect each other.
- Trapezium: One pair of parallel sides; area = ½ × (a + b) × h.
- Kite: Two pairs of adjacent equal sides; diagonals perpendicular; area = ½ × d₁ × d₂.
