Quadrilateral: Definition, Types, Properties, Examples
Quadrilateral is a four-sided polygon with four angles. Learn about its types, properties, area formulas, and more to understand this fundamental shape in geometry.
Shruti Dutta12 Jan, 2025
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A quadrilateral is a polygon with four sides, four vertices, and four angles. It is one of the most basic shapes in geometry and can be classified into various types based on the properties of its sides and angles.
Quadrilaterals are closed figures, meaning their sides connect to form a complete shape. The sum of the interior angles of any quadrilateral is always 360°. Quadrilaterals can have different characteristics, such as parallel sides, equal angles, or equal sides, leading to different categories like squares, rectangles, trapezoids, and rhombuses. Understanding the properties and types of quadrilaterals is fundamental in geometry and its real-world applications.What is a Quadrilateral?
A quadrilateral is a polygon with four sides, four angles, and four vertices. When naming a quadrilateral, following a specific order for the vertices is important. For example, the quadrilateral can be named ABCD, BCDA, ADCB, or DCBA, but not ACBD or DBAC, as these arrangements change the order in which the vertices are connected. In a quadrilateral like ABCD, the four sides are AB, BC, CD, and DA, and there are two diagonals, AC and BD. A quadrilateral is a closed shape formed by joining four points, where no three points are collinear. Simply put, a quadrilateral is a polygon with 4 sides, 4 angles, and 4 vertices. Let's explore more about quadrilaterals, their properties, different types, and examples. [video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Copy-of-Corurious-Jr-Reel-2-Landscape-1-2-1.mp4"][/video]Also Check: Cube
Types of Quadrilaterals
Let's explore the different types of quadrilaterals, each with its own distinct characteristics1. Square
A
square
is a special type of quadrilateral in which all four sides are equal in length and all four interior angles are right angles (90°). This means that every square is a rectangle (because it has four right angles) and a rhombus (because it has four equal sides).
Additionally, a square has two pairs of parallel sides, and its diagonals are both equal in length and perpendicular to each other, bisecting each other at right angles. These properties make the square one of the most symmetrical and regular quadrilaterals.
2. Rectangle
A
rectangle
is a quadrilateral with opposite sides equal in length and all four interior angles measuring 90°. The defining feature of a rectangle is that it has parallel opposite sides, and unlike a square, the adjacent sides can have different lengths.
A rectangle also has two equal diagonals that bisect each other. Because of its right angles and equal diagonals, the rectangle is often used in architecture, design, and geometry.
3. Rhombus
A
rhombus
is a quadrilateral where all four sides are equal in length, similar to a square. However, unlike a square, the interior angles of a rhombus are not necessarily 90°. The opposite sides of a rhombus are parallel, and the opposite angles are equal. T
he diagonals of a rhombus bisect each other at right angles but are not necessarily of equal length. The rhombus is a versatile quadrilateral that is commonly used in tiling patterns and vector-based calculations.
4. Parallelogram
A
parallelogram
is a quadrilateral in which both pairs of opposite sides are parallel. The opposite sides are also equal in length, and the opposite angles are equal. The diagonals of a parallelogram bisect each other but are generally not equal.
The sum of the interior angles of a parallelogram is always 360°. Parallelograms include shapes like rectangles, rhombuses, and squares, making them an important classification in geometry.
5. Trapezium (or Trapezoid in some regions)
A
trapezium
is a quadrilateral with exactly one pair of parallel opposite sides. The parallel sides are called the "bases" of the trapezium, while the non-parallel sides are called the "legs". Unlike other quadrilaterals, trapeziums don’t have equal opposite sides or angles.
Trapeziums are commonly seen in construction and art, where they are used for their simple but useful properties, such as for forming slanted roof shapes or in designs requiring a non-uniform structure.
6. Isosceles Trapezium
An
isosceles trapezium
is a specific type of trapezium where the non-parallel sides, or "legs", are equal in length. This gives the isosceles trapezium a more symmetrical appearance compared to a regular trapezium.
Additionally, the diagonals of an isosceles trapezium are of equal length and intersect at the same angle, which is a distinctive property.
The angles between the parallel sides are also equal. Isosceles trapeziums are frequently used in architectural design and can be found in various structural elements like windows and doorframes.
7. Kite
A
kite
is a quadrilateral that has two pairs of adjacent sides that are equal in length. One pair of opposite angles in a kite are equal, typically the angles between the unequal sides. A notable feature of a kite is that its diagonals intersect at right angles. The longer diagonal bisects the shorter one, creating two congruent triangles.
Kites are commonly found in various fields, including in the design of certain structures and in the recreational flying of kites. They are also notable for their use in certain mathematical problems related to symmetry and area calculation.
Also Check: Isosceles Triangle
Properties of Quadrilateral
Each type of quadrilateral has unique properties, but certain characteristics are common to all quadrilaterals. These are as follows:- All quadrilaterals have four sides.
- All quadrilaterals have four vertices.
- All quadrilaterals have two diagonals.
Square
A square is a quadrilateral with four equal sides and four right angles. Key Properties :- All four sides are equal in length: 𝐴𝐵=𝐵𝐶=𝐶𝐷=𝐷𝐴
- All interior angles are right angles: ∠A=∠B=∠C=∠D=90°.
- It has two pairs of parallel sides: 𝐴AB∥DC and AD∥BC.
- The diagonals are equal in length:AC=BD.
Rectangle
A rectangle is a quadrilateral with opposite sides equal and all interior angles being 90°. Key Properties :- It has two pairs of parallel sides: AB∥DC and AD∥BC.
- All interior angles are 90°: ∠A=∠B=∠C=∠D=90°.
- The opposite sides are equal: AB=DC and AD=BC.
- The diagonals are equal in length: AC=BD, and they bisect each other.
Parallelogram
A parallelogram is a quadrilateral where the opposite sides are parallel. Key Properties: It has two pairs of parallel sides: PQ∥RT and PR∥QT- The opposite sides are equal: PQ=RT and PR=QT.
- The opposite angles are equal: ∠P=∠T and ∠Q=∠R.
Trapezium
A trapezium is a quadrilateral with one pair of opposite sides parallel. Key Properties :- The parallel sides are called the "bases" of the trapezium: EF and GH.
- The non-parallel sides are the "legs": EG and FH.
- In a special type of trapezium called an isosceles trapezium, the non-parallel sides are equal, and the diagonals are also equal in length.
Also Check: Quadratic Equation
Rhombus
A rhombus is a quadrilateral with four equal sides. Key Properties : It has two pairs of parallel sides: EH∥FG and EF∥HG.- All four sides are equal: EH=HG=GF=FE.
- The opposite angles are equal: ∠E=∠G and ∠H=∠F.
- The diagonals are perpendicular and bisect each other: EG⊥HF.
Kite
A kite is a quadrilateral where two pairs of adjacent sides are equal. Key Properties :- It has two pairs of equal adjacent sides: AB=BC and CD=DA.
- One pair of opposite angles (obtuse angles) equals ∠A=∠C.
- The diagonals are perpendicular: AC⊥BD.
- The longer diagonal bisects the shorter diagonal.
- Each quadrilateral has its own distinct set of properties, but they all share common features, such as having four sides and a total interior angle sum of 360°.
Adjacent Sides of a Quadrilateral
Adjacent sides of a quadrilateral are the two sides that share a common endpoint or vertex. In the given example, the quadrilateral has four pairs of adjacent sides:- (AB, BC)
- (BC, CD)
- (CD, DA)
- (DA, AB)
Opposite Sides of a Quadrilateral
Opposite sides of a quadrilateral are the sides that do not share a common endpoint. In the given diagram, there are two pairs of opposite sides:- (DA, BC)
- (CD, AB)
Adjacent Angles of a Quadrilateral
Adjacent angles of a quadrilateral are the angles that share a common side. In the given quadrilateral, there are four pairs of adjacent angles:- (∠D, ∠A)
- (∠A, ∠B)
- (∠B, ∠C)
- (∠C, ∠D)
Opposite Angles of a Quadrilateral
Opposite angles of a quadrilateral are the angles that are not adjacent to each other. In the example, the two pairs of opposite angles are:- (∠A, ∠C)
- (∠B, ∠D)
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Quadrilateral FAQs
What is a cyclic quadrilateral?
A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle, meaning all its vertices lie on the circle. The sum of the opposite angles in a cyclic quadrilateral is 180 degrees.
What is a quadrilateral?
A quadrilateral is a polygon with four sides, four angles, and four vertices. The sum of its interior angles is always 360 degrees.
What are the types of quadrilaterals?
Some common types of quadrilaterals include squares, rectangles, parallelograms, rhombuses, trapezoids, and kites. Each type has unique properties based on the lengths of sides and angles.
Is 360 a quadrilateral?
No, 360 is not a quadrilateral. It is a number, but a quadrilateral refers to a polygon with four sides and four angles.
What is the sum of the angles in a quadrilateral?
The sum of the interior angles of a quadrilateral is always 360°. This is a fundamental property of quadrilaterals.
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