Quadrilateral
A polygon having four sides, four vertices, and four angles is called a quadrilateral. Quadrilaterals, one of the most basic shapes in geometry, are essential to studying mathematics, particularly when it comes to comprehending the characteristics of shapes and how they are used in everyday situations. Quadrilaterals are used in many different contexts, from commonplace items to architectural designs, which makes them crucial to understand. The various kinds of quadrilaterals, their characteristics, angle sums, diagonals, and uses will all be covered in this tutorial.
Table of Contents
- What is a Quadrilateral?
- Types of Quadrilaterals
- Properties of Quadrilaterals
- Angle Sum Property of Quadrilaterals
- Diagonals of Quadrilaterals
- Area and Perimeter of Quadrilaterals
- Applications of Quadrilaterals
- Key Formulas for Quadrilaterals
- Practice Problems
- Conclusion
- FAQs
What is a Quadrilateral?
A polygon with four sides, four vertices, and four interior angles is called a quadrilateral. Any quadrilateral's interior angles add up to 360 degrees. Quadrilaterals can be further classified according to the properties of their angles and sides. They are widely used in many disciplines, including architecture, engineering, and design, and serve as the foundation for comprehending more complicated polygons.
Types of Quadrilaterals
Quadrilaterals come in a variety of forms, each having special characteristics that set them apart. The quadrilaterals that are most frequently found are:
Parallelogram
A quadrilateral with opposite sides that are equal in length and parallel is called a parallelogram. Rectangles and rhombuses are examples of other quadrilaterals that are derived from this basic form.
The parallelogram's characteristics
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The lengths of the opposing sides are equal.
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Angles that are opposite are equal.
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Since they are supplementary, adjacent angles add up to 180°.
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The diagonals cut each other in half, or in two equal halves.
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The following formula is used to determine a parallelogram's area:
Area=Base×Height
Rectangle
A parallelogram with all four angles at 90 degrees is called a rectangle. The opposing sides of a rectangle are parallel and of equal length.
Properties of Rectangle
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Each of the four angles is 90°, or right.
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The opposing sides are parallel and equal.
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The lengths of the diagonals are equal.
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Multiplying the length by the width yields the area:
Area=Length×Width
Square
When all four of a rectangle's sides are the same length, the result is a square. It is a rhombus with equal length sides and a rectangle with four right angles.
Properties of Square
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The length of each side is the same.
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Every angle is 90°, or a right angle.
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The diagonals are the same length and cut each other at right angles.
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The length of one side of a square is squared to determine its area:
Area=Side2
Rhombus
A parallelogram with four equal-length sides is called a rhombus. The angles in a rhombus are not always 90° like those in a square.
Properties of Rhombus
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The length of each side is the same.
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Angles that are opposite are equal.
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Although they are not equal in length, the diagonals cut each other at right angles.
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The following formula is used to determine a rhombus's area:
Trapezium (Trapezoid)
A quadrilateral with just one pair of opposite sides parallel is called a trapezium, or trapezoid in American English. The legs are frequently referred to as the nonparallel sides.
Properties of Trapezium
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There is one parallel pair of opposing sides.
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The trapezium has extra angles on the same side.
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The area is computed by multiplying the height by the average of the parallel side lengths:
Kite
A quadrilateral with two neighbouring side pairs of equal length is called a kite. At right angles, the diagonals cross.
Kite properties
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Two adjacent side pairs are equivalent.
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There is only one equal pair of opposing angles.
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At right angles, the diagonals cross.
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The following formula is used to determine a kite's area:
Properties of Quadrilaterals
Although the characteristics of each form of quadrilateral vary, all quadrilaterals share a few general characteristics:
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Total of the internal angles: Any quadrilateral's interior angles add up to 360 degrees.
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A diagonal is a stretch of a line that joins two vertices that are not contiguous. The diagonals may be equal, bisected, or have no specific relationship at all, depending on the type of quadrilateral.
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Sides: Depending on the type of quadrilateral, its four sides may have equal or differing lengths.
Angle Sum Property of Quadrilaterals
Any quadrilateral's interior angles add up to 360°. This characteristic is crucial for resolving quadrilateral angle related issues. The fact that any quadrilateral may be divided into two triangles and that a triangle's angles always add up to 180° is the basis for the angle sum property. Consequently, a quadrilateral's total angles are:
Regardless of whether the quadrilateral is a square, rectangle, parallelogram, or another shape, this fact is true for all of them.
Diagonals of Quadrilaterals
Line segments that join opposite corners make up a quadrilateral's diagonals. The diagonals may have various characteristics depending on the kind of quadrilateral:
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Within a rectangle, the lengths of the diagonals are equal.
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The diagonals in a rhombus are not always equal in length, but they do cut each other at right angles.
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The diagonals in a kite cross at right angles.
Certain quadrilaterals, like the rhombus and kite, can have their areas calculated with the aid of the diagonals.
Area and Perimeter of Quadrilaterals
A quadrilateral's perimeter is just the total of the lengths of its sides:
A+b+c+d=perimeter
where the side lengths are denoted by a, b, c, and d.
A quadrilateral's area is determined by its type:
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Area is equal to Length × Width for squares and rectangles.
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Area for kites and rhombuses is equal to ½ × Diagonal₁ × Diagonal₂.
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½ × (Base₁ + Base₂) × Height is the area for trapeziums.
Using known values, you may compute the area and perimeter using any of these formulas.
Applications of Quadrilaterals
Quadrilaterals are more than simply theoretical ideas in mathematics; they have real world uses. Among the most popular applications for quadrilaterals are:
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Building and Architecture: When creating windows, doors, rooms, and structures, quadrilaterals are essential. For floor plans, squares and rectangles are frequently used.
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Art and Design: Quadrilaterals are utilised to create layouts, patterns, and logos in this field.
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Engineering: Quadrilateral shapes are used as structural elements in many pieces of equipment and automobiles.
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Sports: Quadrilaterals are used to delineate the boundaries of several sports fields, including football and cricket pitches.
Key Formulas for Quadrilaterals
Here’s a quick reference for key formulas:
Practice Problems
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Find the area of a square with side length 4 cm.
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A rectangle has a length of 8 cm and a width of 5 cm. Find its perimeter and area.
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The diagonals of a rhombus are 6 cm and 8 cm. Find its area.
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A trapezium has parallel sides of lengths 6 cm and 10 cm, and a height of 4 cm. Find its area.
Conclusion
In geometry, quadrilaterals are a fundamental idea. They are crucial for resolving a variety of geometric issues and aid in our understanding of a broad variety of shapes. You are prepared to handle any associated difficulties if you are familiar with the various kinds of quadrilaterals, their characteristics, and the area and perimeter formulas.
Related links:
Angles in Geometry - Discover the concept of angles, their types & their significance in basic geometry.
Angles in Shapes - Learn how angles form the foundation of various shapes & understand their properties.
Parallelogram: Dive into the world of parallelograms and discover their unique properties and formulas.
Frequently Asked Questions on Quadrilateral
1. What are the 7 types of quadrilaterals?
The seven types of quadrilaterals are:
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Parallelogram: The opposing sides are the same length and parallel.
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A parallelogram with four right angles is called a rectangle.
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A square is a rectangle with equal sides on all four sides.
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A parallelogram with equal sides is called a rhombus.
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A quadrilateral having one set of opposing sides parallel is called a trapezium (trapezoid).
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A kite is a quadrilateral with two equal pairs of neighbouring sides.
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An irregular quadrilateral is one in which there are no set rules for the angles and sides.
2. What are the 7 properties of a quadrilateral?
Here are seven key properties of quadrilaterals:
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Sum of Interior Angles: A quadrilateral's interior angles always add up to 360°.
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Sides: Depending on the type, a quadrilateral's four sides can be equal or unequal.
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Vertices: The points where the sides converge are its four vertices.
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Diagonals: Two diagonals join opposite vertices in a quadrilateral.
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Angle Relationships: A quadrilateral's angles may be equal, supplemental, or different, depending on its kind.
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Symmetry: There exist lines of symmetry in certain quadrilaterals, such as squares and rectangles.
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Parallelism: The opposing sides of certain quadrilaterals, such as rectangles and parallelograms, are parallel.
3. What is called a quadrilateral?
A polygon with four sides, four angles, and four vertices is called a quadrilateral. Any quadrilateral's interior angles add up to 360 degrees. Based on their side lengths, angle measurements, and symmetry, quadrilaterals can be categorised into a variety of shapes, including squares, rectangles, parallelograms, and more.
4. Is there 360° in a quadrilateral?
Indeed, any quadrilateral's interior angles add up to 360°. Regardless of their kind, all quadrilaterals share this essential characteristic. A polygon with n sides can have its inner angles added up using the following formula:
For a quadrilateral, where n=4n = 4n=4:
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