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A polygon is a plane figure defined by a closed polygonal chain composed of a finite number of straight line segments (or polygonal circuit). A polygon can be specified as a bounding circuit, a bounded plane region, or both. In this article we are going to discuss a special type of polygon which has four sides.
Definition of a quadrilateral:-
A quadrilateral is a type of polygon with four sides, four vertices, and four angles that is a sort of polygon. It is made up of four non-collinear points that are met together. The sum of all four angles of a quadrilateral always becomes 360°.
The word quadrilateral comes from the Latin words ‘Quadra’ (means four) and ‘Latus’ (means sides). It need not be necessary that all the sides of the quadrilateral are the same. As a result, numerous sorts of quadrilaterals can be created based on sides and angles.
Few properties of a quadrilateral: –
Every quadrilateral has four vertices and four sides, each of which contains four angles.
It is an interior angle that adds up to 360 degrees.
In general, a quadrilateral has its sides of varying lengths and angles of varying degrees. Squares, rectangles, parallelograms, and other quadrilaterals with equal sides and angles are particular sorts of quadrilaterals.
Types of Quadrilateral: –
Another approach to categories quadrilaterals is as follows:
Convex Quadrilaterals:
If a quadrilateral’s diagonals are both completely contained within a figure, then it is known as convex quadrilateral.
Concave Quadrilateral:
If the quadrilateral has at least one diagonal that is partially or completely outside the figure, then the figure is known as concave quadrilateral.
Intersecting Quadrilaterals:
Intersecting quadrilaterals are not the same as basic quadrilaterals with non-adjacent sides intersecting. Self-intersecting or crossed quadrilaterals are these types of quadrilaterals.
Convex, concave, and intersecting quadrilaterals are shown below.
Area of a quadrilateral: –
The formula for the area of a quadrilateral can be found in a variety of ways, including breaking the quadrilateral into two triangles, Heron’s formula, and using the quadrilateral’s sides. Let us have a closer look at each of these methods in detail:
By Dividing into Two Triangles:-
Now let us consider a quadrilateral ABCD with varied
(unequal) lengths, and develop a formula for
its area.
Now the quadrilateral can be viewed as a
composite of two triangles, with the diagonal
BC serves as the common basis.
The heights of triangles ABC and BCD are h1 and h2, respectively.
The area of the given quadrilateral PQRS is equal to the sum of the areas of triangles PSR and PQR.
Now, Area of ∆ABC = ½ (BC × AX)
Area of ∆BCD = ½ (BC × DY)
So, area of the quadrilateral ABCD = Area of ∆ABC + Area of ∆BCD
= ½ (BC × AX) + ½ (BC × DY)
= ½ BC (AX + DY)
Thus the area of the quadrilateral is given by the above-mentioned formula.
Area of Quadrilateral Making use of Heron’s Formula: –
If three sides of a triangle are supplied, Heron’s formula is used to find the area of the triangle. To find the area of the quadrilateral, follow the steps outlined.
Step 1: Using a diagonal with a specified length, divide the quadrilateral into two triangles.
Step 2: To find the area of a quadrilateral, use Heron’s formula for each triangle.
[If the sides of a triangle are a, b, and c, Heron’s formula for finding the area of a triangle is
Area of triangles = √{s(s-a)(s-b)(s-c)}
Where “s” is the triangle’s semi-perimeter, which is equal to (a+b+c)/2.]
Step 3: To calculate the area of a quadrilateral, sum the areas of two triangles.
Area of some types of quadrilaterals: –
The following are the area formulas for various quadrilaterals: square, rectangle, rhombus, kite, parallelogram, and trapezium:
The area of a square = a² sq. units.
Where “a” is the length of the side of the given square.
Rectangle Area = L×B sq. units
Where the length and width of a rectangle are denoted by “L” and “B,” respectively.
Parallelogram Area = B×H sq. units
Where the base length and height of the parallelogram are “B” and “H” respectively.
Rhombus area = ½ d1d2 sq. units.
Where the two diagonals of the rhombus are “d1” and “d2.”
Kite area = ½ p × q sq. units.
The kite’s two diagonals are denoted by “p” and “q.”
Trapezium Area = ½(a+b)h sq. units
Where the side lengths of parallel sides are “a” and “b,” while the trapezium’s height is “h.”
Conclusion:-
The region engrossed by the quadrilateral’s four sides is called the quadrilateral’s area. The term area is one that we are already familiar with. It is defined as the area contained within a flat object’s or figure’s boundary. The measurement is done in square units, with square meters being the usual unit (m²). As we all know, the quadrilateral is a polygon which has four sides. A square, rectangle, rhombus, kite, parallelogram, and trapezium are all examples of quadrilaterals.
