What is a Quadrilateral?

Quadrilateral is a shape with four sides, four corners, and four angles. No matter what type of quadrilateral it is, the total of all its inside angles always adds up to 360 degrees. Examples of quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapezoids, and kites. Each type has its special features and properties but they all are four-sided figures.

• In simple language, A quadrilateral is defined as a polygon having 4 sides, 4 angles, and 4 vertices. The quadrilateral is a type of polygon in which the sides are defined in a proper pattern. 

For example, in the diagram above , the quadrilateral can be defined as ABCD, ADCB, BCDA, CDAB, etc. It cannot be defined as ACBD or BDAC. Here, the quadrilateral’s sides are AB, BC, CD, and DA, and the diagonals are AC and BD.

Basic Properties of Quadrilateral 

The properties of a quadrilateral are:

• It has 4 sides.
• It has 4 vertices.
• It has 4 angles.
• It has 2 diagonals.
• The sum of its internal angles is 360°.
• The sum of its exterior angles is 360°.

Types of Quadrilateral

Based on their properties, quadrilaterals are divided into two major types:

1. Convex quadrilaterals and
2. Concave quadrilaterals.

Concave Quadrilateral

Quadrilaterals that have one interior angle greater than 180° and one diagonal lies outside the quadrilateral are called concave quadrilaterals.

One of the examples of a concave quadrilateral is a Dart. It is a quadrilateral with bilateral symmetry like a kite, but with a reflex interior angle. 

Here, in the image given below, one of the interior angles of the quadrilateral is 210°, which is greater than 180°. Therefore,  the quadrilateral is a concave quadrilateral.

 Convex Quadrilateral

Quadrilaterals that have all four interior angles less than 180° are called concave quadrilaterals. 

There are various types of Convex Quadrilaterals, which are :

1. Trapezium
2. Kite
3. Parallelogram
4. Rectangle
5. Rhombus
6. Square

Quadrilateral Shapes and their properties

A convex quadrilateral is a four sided polygon that has interior angles that measure less than 180 degrees each. The diagonals are contained entirely inside of these quadrilaterals. Convex quadrilaterals can be classified into several sub-categories based on their sides and angles.

Trapezium

A trapezium is a quadrilateral that has one pair of opposite sides parallel. In a regular trapezium, non-parallel sides are equal, and its base angles are equal.

Properties of Trapezium

Here are two important properties of a trapezium:

• The sides of the trapezium that are parallel to each other are known as the bases of trapezium. In the above image, AB and CD are the base of the trapezium.
• The sides of the trapezium that are non-parallel are called the legs. In the above image, AD and BC are the legs.

Area of Trapezium

The area of trapezium is 1/2 × Sum of parallel sides × Distance between them: 

Area =1/2 ​× (a+b) × (h)

Where the parallel sides (a and b) are the two sides of the trapezium that are parallel to each other, and the height (h) is the perpendicular distance between these parallel sides.

Perimeter of Trapezium

The perimeter of a trapezium is the sum of the lengths of all its four sides.

If the trapezium has side lengths a, b, c, and d, then the perimeter is:

Perimeter = a+b+c+d

Kite

Kite has two pairs of equal adjacent sides and one pair of opposite angles equal. Diagonals of kites intersect perpendicularly.

Properties of Kite

Let’s discuss some of the properties of a kite.

A kite has two pairs of equal adjacent sides. For example, AC = BC and AD = BD.

The interior opposite angles that are obtuse are equal; here, ∠A = ∠B.

The diagonals of a kite are perpendicular to each other; here, AB is perpendicular to CD.

The longer diagonal of the kite bisects the shorter diagonal. Here, CD bisects AB.

Area of a Kite

The area of a kite is calculated using the formula:

Area = ½ (d1 x d2)

where, d1 and d2 represent the lengths of the diagonals of the kite.

It’s important to note that this formula works regardless of whether d1 is longer or shorter than d2.

Perimeter of a Kite

The perimeter of a kite, which is the total length of all its sides added together, is calculated using the formula:

Perimeter = 2(a + b)

where, a and b represent the lengths of the two pairs of equal sides of the kite.

Parallelogram

Parallelogram is a quadrilateral whose opposite sides are equal and parallel, Opposite angles of a Parallelogram are equal, and its diagonals bisect each other.

Properties of Parallelogram

The properties of a Parallelogram are:

• The opposite sides of a parallelogram are parallel and equal. In the above example, AB and CD are parallel and equal, and AC and BD are parallel and equal.
• The opposite angles in a parallelogram are equal. ∠A = ∠D and ∠B = ∠C.
• The diagonals of a parallelogram bisect each other.

Area of a Parallelogram

The area of a parallelogram is calculated as the product of its base and height. The formula is given as:

Area of Parallelogram ​= base × height

Where the base is any one of the parallel sides, and the height is the perpendicular distance between the base and its opposite side.

Perimeter of a Parallelogram

The perimeter of a parallelogram is the sum of the lengths of all its sides. Since opposite sides of a parallelogram are equal in length, the perimeter is calculated as:

Perimeter of Parallelogram​ = 2 × (a+b)

Where, a and b are the adjacent sides of a parallelogram.

Rectangle 

Rectangle is a quadrilateral whose opposite sides are equal and parallel and all the interior angles equal to 90°. 

Rectangle Properties

These are some of the important properties of rectangle:

• The opposite sides of a rectangle are parallel and equal. In the above example, AB and CD are parallel and equal, and AC and BD are parallel and equal.
• All 4 angles of a rectangle are equal and are equal to 90°. ∠A = ∠B = ∠C = ∠D = 90°.
• The diagonals of a rectangle bisect each other and the diagonals of a rectangle are equal, here, AD = BC.

Area of a Rectangle

The area of a rectangle is calculated as the product of its length and width. The formula is given as:

Area of Rectangle​ = length × width

Where the length is the longer side and the width is the shorter side of the rectangle.

Perimeter of a Rectangle

The perimeter of a rectangle is the sum of the lengths of all its sides. Since opposite sides of a rectangle are equal in length, the perimeter is calculated as:

Perimeter of Rectangle​ = 2 × (length + width)

Rhombus 

Rhombus is a quadrilateral that has all sides equal and opposite sides parallel. Opposite angles of a rhombus are equal, and diagonals of the Rhombus bisect each other perpendicularly. 

• Note all rhombus are parallelograms, but the reverse of this is not true. 

Properties of Rhombus

Here are some of the key properties of a Rhombus:

• All 4 sides of a rhombus are equal. AB = BC = CD = AD.
• The opposite sides of a rhombus are parallel and equal. In the image above, AB is parallel to CD and AD is parallel to BC.
• The diagonals of a rhombus Bisect each other at 90°.

Area of a Rhombus

The area of a rhombus can be calculated by multiplying the lengths of its two diagonals and dividing by 2. The formula is:

Area of Rhombus ​= 1/2 ​× (diagonal1 × diagonal2​)

Where, diagonal 1 and diagonal 2​ are the lengths of the two diagonals.

Perimeter of a Rhombus

The perimeter of a rhombus is the sum of the lengths of all its four sides. Since all sides of a rhombus are equal in length, the perimeter can be calculated as:

Perimeter of Rhombus​ = 4 × side

Square 

A quadrilateral that has all sides equal and opposite sides parallel and all interior angles equal to 90° is called a Diagonals of squares bisecting each other perpendicularly. Note that all squares are rhombus but not vice-versa.  

Properties of Square

The properties of a square are:

• All four sides of a square are equal to each other.
• The interior angles of a square are 90°.
• The diagonal of a square bisects each other at 90°.
• The opposite sides are parallel, and the adjacent sides are perpendicular in a square.

Area of a Square

The area of a square is calculated by squaring the length of one of its sides. The formula is:

Area of Square​ = side2

Where side is the length of one side of the square.

Perimeter of a Square

The perimeter of a square is the sum of the lengths of all its sides. Since all sides of a square are equal in length, the perimeter can be calculated as:

Perimeter of Square​ = 4 × side

Applications of Quadrilaterals

Architecture and Construction – 

• Building Design: Architects use quadrilaterals extensively in designing buildings. Rectangles and squares, both types of quadrilaterals, are common in floor plans and building structures due to their symmetry and stability. The floor plans, walls, windows, and doors of buildings often involve quadrilateral shapes.
• Roofing Design: Trapezoids, a type of quadrilateral, are used in designing roofs. The slanted sides of trapezoids make them ideal for shedding water and snow efficiently.

Geometry in Art and Design

• Graphic Design: Artists and Graphic designers use quadrilaterals in logo design, poster layouts, and digital art. The symmetry of rectangles and squares creates visually appealing compositions. 

Engineering

• Bridges and roads: Engineers use quadrilaterals in the design and construction of bridges and roads. The support structures, such as trusses and beams, are often based on quadrilateral shapes to distribute weight and provide stability. 

Explanation of Interactive simulation 

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This interactive simulation on quadrilaterals is designed to help users explore and understand the properties of various types of quadrilaterals by manipulating their shapes in real-time. By engaging with the simulation, students can visualize how changes to the vertices, sides, and angles affect the overall classification and characteristics of quadrilaterals. 

Simulation Features

1. Interactive Shape Manipulation

Vertices Adjustment: Users can click and drag the vertices of a quadrilateral, changing its shape dynamically. This feature helps in visualizing how the shape morphs from one type of quadrilateral to another, such as from a square to a rectangle or parallelogram.

Side Lengths: As the vertices are moved, the simulation automatically recalculates and displays the lengths of the sides. This visual feedback helps users see the relationships between the sides and understand properties like equal side lengths in squares and rhombuses.

2. Angle Measurement and Classification

Real -Time Angle Updates: The internal angles of the quadrilateral are displayed and updated in real-time as the shape is manipulated. Users can see how these angles interact and change, maintaining the constant sum of 360 degrees.

Angle-Based Classification: The simulation can classify the quadrilateral based on its angles, such as identifying right-angled quadrilaterals (rectangles) or those with all angles equal (squares).

 3. Quadrilateral Classification

Type Identification: The simulation highlights and identifies the specific type of quadrilateral being formed, whether it’s a rectangle, square, rhombus, parallelogram, trapezium, or kite. This classification helps users understand the defining properties of each type.

Property Verification: Users can verify the properties of each quadrilateral by manipulating the shape to see if it meets the criteria, such as equal opposite sides in parallelograms or perpendicular diagonals in rhombuses.