Types of Quadrilaterals can be daunting tasks to understand as there are so many of them with each having different properties and all being different from each other. All the different types of quadrilaterals have very distinct shapes and properties which help us identify these quadrilaterals. As we know, quadrilaterals are polygons with four sides, four interior angles, and four vertices. With its unique sets of properties and characteristics, there can various different quadrilaterals such as square, rectangle, parallelogram, rhombus, trapezium, and kite.

Understanding these different types of quadrilaterals help us understand the fundamentals of geometry and also helps us deal with further advanced topics in mathematics such as trigonometry and calculus. In this article, we will explore the different types of quadrilaterals, their properties, and how they differ from one another. So, let’s dive into the world of different types of quadrilaterals.

## What are Quadrilaterals?

The word “Quadrilateral” is a combination of two Latin words ‘Quadri‘ means a variant of four, and ‘latus‘ means side, which has a combined meaning of “four sides”. As we already discussed in the introduction of the article Quadrilaterals can be defined as, types of polygons that have four sides, four vertices, and four angles along with a pair of diagonals. Other than that, the Sum of the interior angles of quadrilaterals is 360°.

### What are the Types of Quadrilaterals?

As there are many types of quadrilaterals, with each type having unique sets of properties and characteristics. The most common types of quadrilaterals which we will study in this article are:

- Square
- Rectangle
- Rhombus
- Parallelogram
- Trapezium
- Kite

## Square

A quadrilateral that has all sides equal and opposite sides parallel and all interior angles equal to 90° is called a Square. The diagram of the square is represented in the following illustration:

### Properties of a Square

Some properties of Square are as follows:

- All the sides of a square are equal to each other.
- All the angles of a square are equal and of 90° each.
- Opposite Sides are parallel to each other.
- Diagonals are perpendicular bisectors of each other and are equal.
- The perimeter of a square is equal to the sum of the lengths of all its sides i.e., 4a (where a is the side of the square).
- The area of a square is equal to the length of one side squared i.e., a2 (where a is the side of the square).

Note: All squares are rhombuses but not all rhombuses are squares.

## Rectangle

A rectangle is a quadrilateral in which there are two pair of equal and parallel sides with all the interior angle right angle, i.e. 90°. The following illustration shows the diagram of a general rectangle.

### Properties of a Rectangle

Properties of a rectangle are similar to those of a parallelogram:

- Opposite Sides are parallel to each other.
- Opposite Sides of a rectangle are equal.
- Diagonals bisect each other.
- Diagonals of the rectangle are equal.
- Each interior angle of a rectangle is equal i.e., 90°
- The perimeter of a rectangle is equal to the sum of the lengths of all its sides i.e., 2(l + b) (where l and b are the length and breadth of rectangle respectively).
- The area of a rectangle is equal to the product of its length and width i.e., l × b (where l and b are the length and breadth of the rectangle respectively).

Note: All rectangles are parallelogram, but all parallelograms are not rectangle.

## Rhombus

A quadrilateral that has all sides equal and opposite sides parallel is called Rhombus. The following illustration shows the general diagram of Rhombus:

### Properties of a Rhombus

Properties of a rhombus are similar to those of a parallelogram:

- All the sides of a rhombus are equal to each other.
- Opposite Sides are parallel to each other.
- Opposite angles of a rhombus are equal.
- The diagonals of a rhombus bisect each other.
- The diagonals of a rhombus are perpendicular bisectors of each other.
- The area of a rhombus can be calculated using the formula A = 1/2(d1×d2), (where d1 and d2 are the lengths of the diagonals.)

Note: Rhombus is a parallelogram with all side equal, but parallelogram is not rhombus.

## Parallelogram

A parallelogram is a special type of quadrilateral whose opposite sides are equal and parallel.

### Properties of Parallelogram

There are different properties of parallelograms.

- Opposite sides of the Parallelogram are parallel and equal in length.
- Opposite angles of the Parallelogram are also equal.
- Any two consecutive or adjacent angle in a parallelogram is supplementary i.e., the sum of any two adjacent angles are 180°.
- Diagonals of Parallelogram bisect each other.
- A diagonal of a parallelogram divides it into two congruent triangles.

## Trapezium

A trapezium is a quadrilateral that has one pair of opposite sides parallel. The following illustration shows the general diagram of a trapezium:

### Properties of Trapezium

The different properties of the trapezium are as follows:

- The parallel sides of a trapezium are called bases, and the non-parallel sides are called legs.
- The angles on the same side of the leg are called adjacent angles, and they add up to 180 degrees.
- The diagonals of a trapezium intersect each other.
- The length of the diagonal can be calculated using the formula , (where a and b are the lengths of the bases and h is the distance between the bases.)
- The area of a trapezium can be calculated using the formula A = 1/2(a+b)h, (where a and b are the lengths of the bases and h is the distance between the bases.)

## Kite

A kite is a quadrilateral that has two pairs of adjacent sides that are equal in length. The following illustration shows the general diagram of a kite:

### Properties of Kite

The different properties of the quadrilateral kite are as follows:

- Kite has two pairs of adjacent sides which are equal in length.
- The diagonals of a kite are perpendicular to each other.
- Diagonals of kite bisect each other.
- The longer diagonal of a kite bisects the angle between the unequal sides, while the shorter diagonal bisects the angle between the equal sides.
- The area of a kite can be calculated using the formula A = 1/2(d1×d2), (where d1 and d2 are the lengths of the diagonals).

## Concave and Convex Quadrilaterals

Based on the value of internal angles, quadrilaterals or any other polygon can be classified into concave and convex.

### Concave Quadrilaterals

Those quadrilaterals which have at least one of the interior angles greater than 180° are called concave quadrilaterals. We can also define concave quadrilaterals as those quadrilaterals for which any one of its diagonals lies outside the area bounded by the sides of the quadrilateral.

### Convex Quadrilaterals

Those quadrilaterals in which all the interior angles are less than 180° are called convex quadrilaterals. We can also define convex quadrilaterals as those quadrilaterals for which none of its diagonals lies outside the area bounded by the sides of the quadrilateral.

## Sample Problems on Types of Quadrilaterals

Problem 1: All rhombus are squares or all squares are rhombus. Which of these statements is correct and why?

Solution:

Square and rhombus both have all sides equal, but a rhombus is called square if each of its angle is 900. So all squares can be called rhombus, but converse is not true.

Problem 2: In the figure ROPE is a square. Show that diagonals are equal.

Solution:

In Δ REP and Δ OEP

RE = OP (sides of square

∠E = ∠P (each 90°)

EP = EP (common)

Therefore, triangles are congruent by SAS criteria.

Therefore, RP = OE (c.p.c.t)

Therefore, diagonals of square are equal.

Problem 3: In rectangle ABCD, AO = 5cm. Find the length of diagonal BD. Also, find the perimeter of the rectangle if AB = 8cm and AD = 6cm.

Solution:

AO = OC = 5cm (diagonals bisect each other)

Therefore, AC = 10cm

BD = AC =10cm (diagonals of rectangle are equal)

Perimeter = AB + BC + CD + DA

= 8 + 6 +8 +6 (opposite sides are equal)

= 28cm

Problem 4: In rectangle ABCD, ∠ABD = 3x – 7 and ∠CBA = 6x – 2. Find the value of x.

Solution:

Each angle of rectangle is 90°

Therefore,

∠ABD + ∠CBA = 90°

3x – 7 + 6x – 2 = 90

9x – 9 = 90

9x = 99

x = 11

Problem 5: In rectangle ABCD AO = 2x – 10 cm, OB = x + 4 cm. Find the length of diagonal BD

Solution:

In rectangle diagonals bisect each other and are equal.

Therefore, AO = OB

2x – 10 = x + 4

x = 14

OB = 14 + 4 = 18 cm

OD = 18 cm (as diagonals bisect each other)

Therefore, BD = 36 cm

Problem 6: Diagonals of the rhombus are 24cm and 10cm. Find the side of the rhombus.

Solution:

AC = 24cm

BD = 10cm

Therefore, AO = 12cm and OB = 5cm (diagonals bisect each other)

In right-angled triangle AOB, (diagonals of rhombus are perpendicular)

AB2 = OA2 + OB2

AB2 = 122 + 52

AB2 = 144 + 25

AB2 = 169

AB = 13cm

Therefore, side of rhombus is 13cm.

Problem 7: In a rhombus one of the diagonals is equal to a side of the rhombus. Find the angles of the Rhombus.

Solution:

In rhombus PQRS PR = PQ (given)

Therefore, PQ = QR = RS = SP = PR (as all side of rhombus are equal)

In triangle PQR

PQ = QR = PR

Therefore, it is an equilateral triangle.

∠QPR = ∠Q = ∠QRP = 60°

>||ly ∠SPR = ∠S = ∠PRS = 60°

Therefore, angles of rhombus are ∠P = 120°, ∠Q = 60°, ∠R = 120°, ∠S = 60°

Problem 8: Derive the formula for the area of the rhombus i.e., 1/2 × Product of Diagonals.

Solution:

As diagonals of rhombus bisect each other at right angle.

In rhombus ABCD

Area of Triangle ABD = 1/2 × BD ×AO (As area of triangle = 1/2 × base× height) . . . (1)

Area of Triangle BCD = 1/2 × BD × CO . . . (2)

Thus, Area of rhombus = Area of triangle ABD + area of triangle BCD

⇒ Area of rhombus = 1/2 × BD × AO + 1/2 × BD × CO

⇒ Area of rhombus = 1/2 × BD (AO + CO)

⇒ Area of rhombus = 1/2 × BD × AC (AE + CE = AC)

Therefore, area of rhombus = 1/2 × Product of Diagonals

## FAQs on Types of Quadrilaterals

### Q1: Define Quadrilateral.

Answer:

A quadrilateral is a polygon with four sides and four vertices.

### Q2: What are the Different Types of Quadrilaterals?

Answer:

The different types of quadrilaterals are:

- Square
- Rectangle
- Rhombus
- Parallelogram
- Trapezium
- Kite

### Q3: What is a Rectangle?

Answer:

A rectangle is a quadrilateral with four right angles and opposite sides of equal length.

### Q4: What is a Square?

Answer:

A square is a quadrilateral with four right angles and four sides of equal length.

### Q5: What is a Parallelogram?

Answer:

A parallelogram is a quadrilateral with opposite sides parallel and equal in length.

### Q6: What is a Rhombus?

Answer:

A rhombus is a quadrilateral with four sides of equal length.

### Q7: What is a Trapezoid?

Answer:

A trapezoid is a quadrilateral with one pair of opposite sides parallel.

### Q8: What is a Kite?

Answer:

A kite is a quadrilateral with two pairs of adjacent sides that are equal in length, and the diagonals intersect at right angles.

### Q9: Can a Rectangle also be a Square?

Answer:

Yes, a rectangle can also be a square if all four sides are of equal length.

### Q10: Can a Square also be a Rectangle?

Answer:

Yes, a square can also be a rectangle if all four angles are right angles.

### Q11: Can a Rhombus also be a Square?

Answer:

Yes, a rhombus can also be a square if all four angles are right angles.

### Q12: Can a Square also be a Rhombus?

Answer:

Yes, a square can also be a rhombus if all four sides are of equal length.

### Q13: How many Diagonals does a Quadrilateral have?

Answer:

A quadrilateral has two diagonals.