Types of Quadrilaterals: Complete Guide
A quadrilateral is a polygon with four sides (or edges) and four vertices (or corners). The word "quadrilateral" comes from the Latin words "quadri" (meaning "four") and "latus" (meaning "side").
Hierarchy of Quadrilaterals
Types of Quadrilaterals
Parallelogram
- Opposite sides are parallel and equal in length
- Opposite angles are equal
- Consecutive angles are supplementary (sum to 180°)
- Diagonals bisect each other
Area = base × height
Perimeter = 2(a + b)
where a and b are the lengths of adjacent sides
Rectangle
- All angles are 90° (right angles)
- Opposite sides are parallel and equal
- Diagonals are equal and bisect each other
- Is a special type of parallelogram
Area = length × width
Perimeter = 2(length + width)
Diagonal = √(length² + width²)
Square
- All sides are equal in length
- All angles are 90° (right angles)
- Diagonals are equal, bisect each other, and intersect at 90°
- Is both a rhombus and a rectangle
Area = side²
Perimeter = 4 × side
Diagonal = side × √2
Rhombus
- All sides are equal in length
- Opposite sides are parallel
- Opposite angles are equal
- Diagonals bisect each other at 90°
- Is a special type of parallelogram
Area = (d₁ × d₂) ÷ 2
where d₁ and d₂ are the lengths of the diagonals
Perimeter = 4 × side
Trapezoid (Trapezium)
- Exactly one pair of opposite sides are parallel
- The parallel sides are called bases
- The non-parallel sides are called legs
Area = [(a + c) × h] ÷ 2
where a and c are the lengths of the parallel sides
and h is the height (distance between parallel sides)
Perimeter = sum of all sides
Isosceles Trapezoid
- One pair of opposite sides are parallel
- The non-parallel sides (legs) are equal in length
- Base angles are equal
- Diagonals are equal in length
Area = [(a + c) × h] ÷ 2
where a and c are the lengths of the parallel sides
and h is the height
Kite
- Two pairs of adjacent sides are equal in length
- One diagonal bisects the other diagonal
- One diagonal bisects two opposite angles
- The diagonals are perpendicular
Area = (d₁ × d₂) ÷ 2
where d₁ and d₂ are the lengths of the diagonals
Perimeter = 2(a + b)
where a and b are the lengths of the two different sides
Comparison of Quadrilateral Properties
| Property | Square | Rectangle | Rhombus | Parallelogram | Trapezoid | Kite |
|---|---|---|---|---|---|---|
| All sides equal | Yes | No | Yes | No | No | No |
| All angles equal | Yes | Yes | No | No | No | No |
| Opposite sides parallel | Yes | Yes | Yes | Yes | Partially | No |
| Opposite sides equal | Yes | Yes | Yes | Yes | No | No |
| Diagonals bisect each other | Yes | Yes | Yes | Yes | No | No |
| Diagonals are equal | Yes | Yes | No | No | No* | No |
| Diagonals perpendicular | Yes | No | Yes | No | No | Yes |
* Diagonals are equal in an isosceles trapezoid
Methods for Solving Quadrilateral Problems
Method 1: Using Area Formulas
- Identify the type of quadrilateral
- Select the appropriate area formula based on the shape
- Substitute the known values into the formula
- Solve for the area or any unknown dimensions
Method 2: Using Coordinate Geometry
- Plot the vertices of the quadrilateral on the coordinate plane
- Calculate side lengths using the distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
- Calculate slopes to determine if sides are parallel or perpendicular
- Use the formula for area: A = (1/2)|x₁(y₂ - y₄) + x₂(y₃ - y₁) + x₃(y₄ - y₂) + x₄(y₁ - y₃)|
Method 3: Using Trigonometry
- Use the Law of Cosines to find unknown sides: c² = a² + b² - 2ab·cos(C)
- Use the Law of Sines to find unknown angles: sin(A)/a = sin(B)/b = sin(C)/c
- Calculate area using the formula: A = (1/2)ab·sin(C)
Method 4: Using Diagonals
- Identify the diagonals of the quadrilateral
- For specific shapes like kites or rhombuses, use the formula: A = (d₁ × d₂) ÷ 2
- For general quadrilaterals, divide into triangles and sum their areas
Method 5: Using Properties for Verification
- Check if the shape has properties of a specific quadrilateral (e.g., equal sides, right angles)
- Verify parallel sides by comparing slopes
- Check angle measures (e.g., using angle sum property: sum of angles = 360°)
- Analyze diagonal properties (e.g., perpendicular, bisecting)
Example Problems
Example 1: Finding the Area of a Parallelogram
A parallelogram has a base of 8 cm and a height of 5 cm. Calculate its area.
Solution:
For a parallelogram, Area = base × height
Area = 8 cm × 5 cm = 40 cm²
Example 2: Finding the Perimeter of a Rectangle
A rectangle has a length of 12 cm and a width of 7 cm. Find its perimeter and diagonal length.
Solution:
Perimeter = 2(length + width) = 2(12 + 7) = 2(19) = 38 cm
Diagonal = √(length² + width²) = √(12² + 7²) = √(144 + 49) = √193 ≈ 13.89 cm
Example 3: Determining if a Quadrilateral is a Rhombus
A quadrilateral has sides of lengths 5 cm, 5 cm, 5 cm, and 5 cm. What type of quadrilateral is it?
Solution:
Since all sides are equal (5 cm), this is at least a rhombus.
However, without information about the angles, we cannot determine if it's a square (which would require all angles to be 90°).
Therefore, based on the given information, it is a rhombus.
Example 4: Finding the Area of a Trapezoid
A trapezoid has parallel sides of lengths 10 cm and 16 cm, with a height of 8 cm. Calculate its area.
Solution:
For a trapezoid, Area = [(a + c) × h] ÷ 2
where a and c are the lengths of the parallel sides and h is the height
Area = [(10 + 16) × 8] ÷ 2 = [26 × 8] ÷ 2 = 208 ÷ 2 = 104 cm²
Example 5: Using Coordinate Geometry
Determine the type of quadrilateral with vertices at A(0,0), B(4,0), C(4,3), and D(0,3).
Solution:
Calculate the side lengths:
AB = √[(4-0)² + (0-0)²] = 4
BC = √[(4-4)² + (3-0)²] = 3
CD = √[(0-4)² + (3-3)²] = 4
DA = √[(0-0)² + (0-3)²] = 3
Opposite sides are equal: AB = CD and BC = DA
Check angles: Since the sides form horizontal and vertical lines, all angles are 90°
Therefore, this is a rectangle.