In Euclidean geometry, a quadrilateral is a four-sided 2D figure whose sum of internal angles is 360°. The word quadrilateral is derived from two Latin words ‘quadri’ and ‘latus’ meaning four and side respectively. Therefore, identifying the properties of quadrilaterals is important when trying to distinguish them from other polygons. So, what are the properties of quadrilaterals? There are two properties of quadrilaterals:

- A quadrilateral should be closed shape with 4 sides
- All the internal angles of a quadrilateral sum up to 360°

In this article, you will get an idea about the 5 types of quadrilaterals and get to know about the properties of quadrilaterals.

This is what you’ll read in the article:

- Classifying quadrilaterals – Different types of quadrilaterals
- Rectangle
- Square
- Parallelogram
- Rhombus
- Trapezium/Trapezoid
- Properties of quadrilaterals
- Important quadrilateral formulas
- Quadrilateral questions

The diagram given below shows a quadrilateral ABCD and the sum of its internal angles. All the internal angles sum up to 360°.

Thus, ∠A + ∠B + ∠C + ∠D = 360°

## Classifying quadrilaterals – Different types of quadrilaterals

We can classify quadrilaterals into 5 types on the basis of their shape. We’ll discuss each one of them in this article. These 5 quadrilaterals are:

- Rectangle
- Square
- Parallelogram
- Rhombus
- Trapezium

Let’s discuss each of these 5 quadrilaterals:

## Rectangle

A rectangle is a quadrilateral with four right angles. Thus, all the angles in a rectangle are equal (360°/4 = 90°). Moreover, the opposite sides of a rectangle are parallel and equal, and diagonals bisect each other.

### Properties of rectangles

A rectangle has three properties:

- All the angles of a rectangle are 90°
- Opposite sides of a rectangle are equal and Parallel
- Diagonals of a rectangle bisect each other

### Rectangle formula – Area and perimeter of a rectangle

If the length of the rectangle is L and breadth is B then,

- Area of a rectangle = Length × Breadth or L × B
- Perimeter of rectangle = 2 × (L + B)

## Square

Square is a quadrilateral with four equal sides and angles. It’s also a regular quadrilateral as both its sides and angles are equal. Just like a rectangle, a square has four angles of 90° each. It can also be seen as a rectangle whose two adjacent sides are equal.

### Properties of a square

For a quadrilateral to be a square, it has to have certain properties. Here are the three properties of squares:

- All the angles of a square are 90°
- All sides of a square are equal and parallel to each other
- Diagonals bisect each other perpendicularly

### Square formula – Area and perimeter of a square

If the side of a square is ‘a’ then,

- Area of the square = a × a = a²
- Perimeter of the square = 2 × (a + a) = 4a

## Parallelogram

A parallelogram, as the name suggests, is a simple quadrilateral whose opposite sides are parallel. Thus, it has two pairs of parallel sides. Moreover, the opposite angles in a parallelogram are equal and its diagonals bisect each other i.e., intersect each other at 90°.

### Properties of parallelogram

A quadrilateral satisfying the below-mentioned properties will be classified as a parallelogram. A parallelogram has four properties:

- Opposite angles are equal
- Opposite sides are equal and parallel
- Diagonals bisect each other
- Sum of any two adjacent angles is 180°

### Parallelogram formulas – Area and perimeter of a parallelogram

If the length of a parallelogram is ‘l’, breadth is ‘b’ and height is ‘h’ then:

- Perimeter of parallelogram= 2 × (l + b)
- Area of the parallelogram = l × h

## Rhombus

A rhombus is a quadrilateral whose all four sides are equal in length and opposite sides are parallel to each other. However, the angles are not equal to 90°. A rhombus with right angles would become a square. Another name for rhombus is ‘diamond’ as it looks similar to the diamond suit in playing cards.

### Properties of rhombus

A rhombus is a quadrilateral which has the following four properties:

- Opposite angles are equal
- All sides are equal and, opposite sides are parallel to each other
- Diagonals bisect each other perpendicularly
- Sum of any two adjacent angles is 180°

### Rhombus formulas – Area and perimeter of a rhombus

If the side of a rhombus is a then, perimeter of a rhombus = 4a

If the length of two diagonals of the rhombus is d_{1} and d_{2} then the area of a rhombus = ½ × d_{1} × d_{2}

## Trapezium

A trapezium (called Trapezoid in the US) is a quadrilateral which has only one pair of parallel sides. The parallel sides are referred to as ‘bases’ and the other two sides are called ‘legs’ or lateral sides.

### Properties of Trapezium

A trapezium is a quadrilateral in which the following one property:

- Only one pair of opposite sides are parallel to each other

### Trapezium formulas – Area and perimeter of a trapezium

If the height of a trapezium is ‘*h’ *(as shown in the above diagram) then:

- Perimeter of the trapezium= Sum of lengths of all the sides = AB + BC + CD + DA
- Area of the trapezium = ½ × (Sum of lengths of parallel sides) × h = ½ × (AB + CD) × h

## Properties of quadrilaterals

The below table summarizes all the properties of the quadrilaterals that we have learned so far:

Properties of quadrilaterals | Rectangle | Square | Parallelogram | Rhombus | Trapezium |

All Sides are equal | ✖ | ✔ | ✖ | ✔ | ✖ |

Opposite Sides are equal | ✔ | ✔ | ✔ | ✔ | ✖ |

Opposite Sides are parallel | ✔ | ✔ | ✔ | ✔ | ✔ |

All angles are equal | ✔ | ✔ | ✖ | ✖ | ✖ |

Opposite angles are equal | ✔ | ✔ | ✔ | ✔ | ✖ |

Sum of two adjacent angles is 180 | ✔ | ✔ | ✔ | ✔ | ✖ |

Bisect each other | ✔ | ✔ | ✔ | ✔ | ✖ |

Bisect perpendicularly | ✖ | ✔ | ✖ | ✔ | ✖ |

The below image also summarizes the properties of quadrilaterals:

## Important quadrilateral formulas

The below table summarizes the formulas on area and perimeter of different types of quadrilaterals:

Quadrilateral formulas |
Rectangle |
Square |
Parallelogram |
Rhombus |
Trapezium |

Area |
l × b | a² | l × h | ½ × d1 × d2 | ½ × (Sum of parallel sides) × height |

Perimeter |
2 × (l + b) | 4a | 2 × (l + b) | 4a | Sum of all the sides |

## Quadrilateral questions

Let’s practice the application of properties of quadrilaterals on the following sample questions:

### Question 1

Adam wants to build a fence around his rectangular garden of length 10 meters and width 15 meters. How many meters of the fence he should buy to fence the entire garden?

- 20 meters
- 25 meters
- 30 meters
- 40 meters
- 50 meters

##### Solution

**Step 1: Given**

- Adam has a rectangular garden.
- It has a length of 10 meters and a width of 15 meters.
- He wants to build a fence around it.

**Step 2: To find**

- The length required to build the fence around the entire garden.

**Step 3: Approach and Working out**

The fence can only be built around the outside sides of the garden.

- So, the total length of the fence required= Sum of lengths of all the sides of the garden.
- Since the garden is rectangular, the sum of the length of all the sides is nothing but the perimeter of the garden.
- Perimeter = 2 × (10 + 15) = 50 metres

Hence, the required length of the fence is 50 meters.

Therefore, option E is the correct Answer.

### Question: 2

Steve wants to paint one rectangular shaped wall of his room. The cost to paint the wall is $1.5 per square meter. If the wall is 25 meters long and 18 meters wide, then what is the total cost to paint the wall?

- $ 300
- $ 350
- $ 450
- $ 600
- $ 675

##### Solution

**Step 1: Given**

- Steve wants to paint one wall of his room.
- The wall is 25 meters long and 18 meters wide.
- Cost to paint the wall is $1.5 per square meter.

**Step 2: To find**

- The total cost to paint the wall.

**Step 3: Approach and Working out**

- A wall is painted across its entire area.
- So, if we find the total area of the wall in square meters and multiply it by the cost to paint 1 square meter of the wall then we can the total cost.
- Area of the wall = length × Breadth = 25 metres × 18 metres = 450 square metre
- Total cost to paint the wall = 450 × $1.5 = $675

Hence, the correct answer is option E.

We hope by now you would have learned the different types of quadrilaterals, their properties, and formulas and how to apply these concepts to solve questions on quadrilaterals. The application of quadrilaterals is important to solve geometry questions on the GMAT. If you are planning to take the GMAT, we can help you with high-quality study material which you can access for free by registering here.

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