Quadrilateral
In geometry, a quadrilateral is a closed shape that is formed by joining four points among which any three points are non-collinear. A quadrilateral has 4 sides, 4 angles, and 4 vertices. 'Quadrilateral' is derived from a Latin word, in which, 'Quadra' means four and 'Latus' means sides. All 4 sides of a quadrilateral may or may not be equal.
| 1. | Quadrilateral Definition |
| 2. | Types of Quadrilateral |
| 3. | Properties of Quadrilateral |
| 4. | FAQs on Quadrilaterals |
Quadrilateral Definition
A quadrilateral is a polygon with four sides four angles and four vertices. Whenever we name a quadrilateral, we need to keep in mind the order of the vertices. For example, the following quadrilateral should be named ABCD, BCDA, ADCB, or, DCBA. It cannot be named as ACBD or DBAC, since they change the order of vertices in which a quadrilateral is formed. The following quadrilateral ABCD has four sides: AB, BC, CD, DA, and two diagonals: AC and BD.
Types of Quadrilateral
Although a quadrilateral always has four sides, four angles, and four vertices, the measure of the sides and angles differ. It is to be noted that the sum of the interior angles of a quadrilateral is always equal to 360°. The following table lists the different types of quadrilaterals.
Properties of Quadrilateral
Each of the quadrilateral discussed above has its own properties. Though, there are some properties that are common to all quadrilaterals. They are as follows.
- They have four sides.
- They have four vertices.
- They have two diagonals.
- The sum of all interior angles is 360°.
We will study the other properties of different quadrilaterals in detail. We can identify a quadrilateral by using the following properties of quadrilaterals.
Square
A square is a quadrilateral with four equal sides and four right angles.
A square has:
- Four equal sides
AB = BC = CD = DA - Four right angles
∠ A= ∠B = ∠C = ∠D = 90° - Two pairs of parallel sides
AB ∥ DC and AD ∥ BC - Two equal diagonals
AC = BD - Diagonals that are perpendicular to each other
AC ⊥ BD - Diagonals that bisect each other.
Rectangle
A rectangle is a quadrilateral with two pairs of equal and parallel opposite sides and four right angles.
A rectangle has:
- Two pairs of parallel sides
AB ∥ DC and AD ∥ BC - Four right angles
∠A = ∠B = ∠C = ∠D = 90° - Opposite sides of equal lengths
AB = DC and AD = BC - Two equal diagonals
AC=BD - Diagonals that bisect each other.
Parallelogram
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
A parallelogram has:
- Two pairs of parallel sides
PQ ∥ RT and PR ∥ QT - Opposite sides of equal lengths
PQ = RT and PR = QT - Opposite angles that are equal
∠P = ∠T and ∠Q = ∠R - Two diagonals that bisect each other.
Trapezium
A trapezium is a quadrilateral in which one pair of opposite sides is parallel.
- The sides that are parallel to each other are called bases.
In the above figure, EF and GH are the bases. - The sides that are not parallel to each other are called legs.
In the above figure, EG and FH are legs.
There is nothing special about the sides, angles, or diagonals of a trapezium. But if the two non-parallel opposite sides are of equal length, then it is called an isosceles trapezium. The following quadrilateral XYZW is an isosceles trapezium, in which the legs are equal, i.e., WX = ZY, and the diagonals are also equal, i.e., XZ = WY.
Rhombus
A rhombus is a quadrilateral with four equal sides.
A rhombus has:
- Two pairs of parallel sides
EH ∥ FG and EF ∥ HG - Four equal sides
EH = HG = GF = FE - Opposite angles that are equal
∠E = ∠G and ∠H = ∠F - Diagonals that are perpendicular to each other
EG ⊥ HF - Diagonals that bisect each other.
Kite
A kite is a quadrilateral in which two pairs of adjacent sides are equal.
Observe the following properties of a kite. A kite has:
- Two pairs of equal adjacent sides
AB = BC and CD = DA - One pair of opposite angles (which are obtuse) that are equal
∠A = ∠C - Diagonals that are perpendicular to each other
AC ⊥ BD - A longer diagonal bisects the shorter diagonal.
Think Tank
- Can a kite be called a parallelogram?
- What elements of a trapezium should be changed to make it a parallelogram?
Area of Quadrilaterals
The area of a quadrilateral is the number of unit squares that can be fit into it. The following table lists the formulas for finding the area of quadrilaterals.
☛Topics Related to Quadrilaterals
Check out some interesting articles related to quadrilaterals.
Quadrilateral Examples
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Example 1: Find the value of angle x° in the following figure.
Solution:
We know that the sum of the angles in a quadrilateral is 360°.
This can be written as: x + 67 + 77 + 101 = 360°
x + 245 = 360°
Therefore, x =115° -
Example 3: What is the area of the following quadrilateral (isosceles trapezium)?
Solution:
The area of an isosceles trapezium is calculated using the formula: Area of an isosceles trapezium is (1/2) × (a + b) × h, where 'a' and 'b' are the bases and 'h' is the height of the isosceles trapezium. In this example, the given dimensions are:
a = 16, b = 40 and h = 15
Area = (1/2) × (a + b) × h
=(1/2) × (16 + 40) × 15
= (1/2) × 56 × 15
= 420 square units.
FAQs on Quadrilateral
What is a Quadrilateral?
A quadrilateral is a closed two-dimensional figure that has 4 sides, 4 angles, and 4 vertices. A few examples of quadrilaterals are square, rectangle and trapezium.
What are the Different Types of Quadrilaterals?
There are different types of quadrilaterals that are identified on the basis of their unique properties. For example, square, rectangle, parallelogram, rhombus, kite, trapezium, isosceles trapezium are all categorized under quadrilaterals.
What is the Sum of the Interior Angles in a Quadrilateral?
In any type of quadrilateral, the sum of the interior angles is always equal to 360°. For example, a rectangle is a quadrilateral with each of its interior angles equal to 90° which makes it (90 × 4) = 360°.
What are All the Properties that Quadrilaterals Share in Common?
Though there are different types of quadrilaterals, they share a few properties that are common. They are listed as follows:
- They have four sides.
- They have four vertices.
- They have two diagonals.
- The sum of all interior angles is 360°.
How to Find the Area of a Quadrilateral?
Area is the space that is covered by any shape. A quadrilateral is a figure that has 4 sides and the space occupied by a quadrilateral is called its area. A few examples of quadrilaterals are square and rectangle. The area of a square of side 'a' is calculated by the formula: Area = 'a × a' or a2 and the area of a rectangle whose length is 'l' and width is 'w' is calculated by the formula: Area = 'l × b'. The area of a quadrilateral is always expressed in square units.
How do you Find the Perimeter of a Quadrilateral?
Perimeter is the length of the boundary of any shape. As we know that a quadrilateral has 4 sides, the perimeter of a quadrilateral can be found by adding all the sides of the quadrilateral. For example, if a rectangle has a length of 6 units and a width of 4 units then we use the formula for the perimeter of a rectangle which is: 2(length + width). Substituting the values in the formula: 2 (6 + 4) = 20 units.