Quadrilateral
Quadrilateral
A quadrilateral is a polygon with four sides. The sum of the interior angles of a quadrilateral is 360 degrees. Quadrilaterals can be classified into different types based on their properties. Some common types of quadrilaterals include squares, rectangles, parallelograms, rhombuses, and trapezoids. Squares and rectangles are quadrilaterals with four right angles. Parallelograms are quadrilaterals with opposite sides parallel. Rhombuses are parallelograms with all sides equal in length. Trapezoids are quadrilaterals with at least one pair of parallel sides.
What is a Quadrilateral?
A quadrilateral is a polygon with four sides and four angles. It is a twodimensional shape that can be either convex or concave. A convex quadrilateral has all of its interior angles less than 180 degrees, while a concave quadrilateral has at least one interior angle greater than 180 degrees.
There are many different types of quadrilaterals, including:
 Rectangle: A rectangle is a quadrilateral with four right angles.
 Square: A square is a rectangle with all four sides of equal length.
 Rhombus: A rhombus is a quadrilateral with all four sides of equal length, but its angles are not necessarily right angles.
 Parallelogram: A parallelogram is a quadrilateral with opposite sides parallel and of equal length.
 Trapezoid: A trapezoid is a quadrilateral with only one pair of parallel sides.
 Kite: A kite is a quadrilateral with two pairs of adjacent sides of equal length.
Quadrilaterals are used in many different applications, such as:
 Architecture: Quadrilaterals are used to create the basic shapes of buildings, such as rectangles, squares, and triangles.
 Engineering: Quadrilaterals are used to design and build bridges, roads, and other structures.
 Art: Quadrilaterals are used to create paintings, drawings, and sculptures.
 Science: Quadrilaterals are used to study geometry, physics, and other scientific disciplines.
Here are some examples of quadrilaterals:
 A rectangle is a quadrilateral with four right angles.
 A square is a rectangle with all four sides of equal length.
 A rhombus is a quadrilateral with all four sides of equal length, but its angles are not necessarily right angles.
 A parallelogram is a quadrilateral with opposite sides parallel and of equal length.
 A trapezoid is a quadrilateral with only one pair of parallel sides.
 A kite is a quadrilateral with two pairs of adjacent sides of equal length.
Quadrilaterals are a versatile and important shape that has many different applications in the real world.
Types of Quadrilaterals
Types of Quadrilaterals
A quadrilateral is a polygon with four sides. There are many different types of quadrilaterals, each with its own unique properties. Some of the most common types of quadrilaterals include:
 Parallelogram: A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length.
 Rectangle: A rectangle is a parallelogram with four right angles. The opposite sides of a rectangle are equal in length.
 Square: A square is a rectangle with all four sides equal in length.
 Rhombus: A rhombus is a parallelogram with all four sides equal in length. The angles of a rhombus are not necessarily right angles.
 Trapezoid: A trapezoid is a quadrilateral with only one pair of parallel sides. The opposite sides of a trapezoid are not equal in length.
 Kite: A kite is a quadrilateral with two pairs of adjacent sides that are equal in length. The opposite angles of a kite are equal.
Examples of Quadrilaterals
Here are some examples of quadrilaterals:
 A book is a rectangular prism.
 A window is a square.
 A diamond is a rhombus.
 A kite is a kite.
 A trapezoid is a trapezoid.
Properties of Quadrilaterals
The properties of quadrilaterals can be used to identify different types of quadrilaterals. Some of the properties of quadrilaterals include:
 Side lengths: The side lengths of a quadrilateral can be used to determine if it is a square, rectangle, rhombus, or trapezoid.
 Angles: The angles of a quadrilateral can be used to determine if it is a rectangle, square, or rhombus.
 Diagonals: The diagonals of a quadrilateral can be used to determine if it is a parallelogram or a trapezoid.
Applications of Quadrilaterals
Quadrilaterals are used in a variety of applications, including:
 Architecture: Quadrilaterals are used in the design of buildings, bridges, and other structures.
 Engineering: Quadrilaterals are used in the design of machines, vehicles, and other products.
 Art: Quadrilaterals are used in the creation of paintings, drawings, and other works of art.
Quadrilaterals are a versatile and important part of our world. They are used in a variety of applications, from architecture to engineering to art.
Convex, Concave and Intersecting Quadrilaterals
Convex Quadrilaterals
A convex quadrilateral is a quadrilateral in which all interior angles are less than 180 degrees. In other words, no angle in a convex quadrilateral is pointing inward.
Properties of Convex Quadrilaterals
 The opposite angles of a convex quadrilateral are congruent.
 The diagonals of a convex quadrilateral intersect inside the quadrilateral.
 The sum of the interior angles of a convex quadrilateral is 360 degrees.
Examples of Convex Quadrilaterals
 Square
 Rectangle
 Rhombus
 Trapezoid
Concave Quadrilaterals
A concave quadrilateral is a quadrilateral in which at least one interior angle is greater than 180 degrees. In other words, at least one angle in a concave quadrilateral is pointing inward.
Properties of Concave Quadrilaterals
 The opposite angles of a concave quadrilateral are not congruent.
 The diagonals of a concave quadrilateral intersect outside the quadrilateral.
 The sum of the interior angles of a concave quadrilateral is greater than 360 degrees.
Examples of Concave Quadrilaterals
 Kite
 Dart
 Crescent
Intersecting Quadrilaterals
Intersecting quadrilaterals are quadrilaterals that intersect each other. In other words, the sides of intersecting quadrilaterals cross each other at least once.
Properties of Intersecting Quadrilaterals
 The opposite sides of intersecting quadrilaterals are not parallel.
 The diagonals of intersecting quadrilaterals do not intersect.
 The sum of the interior angles of intersecting quadrilaterals is not a constant.
Examples of Intersecting Quadrilaterals
 Xshape
 Hourglass
 Bow tie
Properties of Quadrilaterals
Properties of Quadrilaterals:
Quadrilaterals are polygons with four sides and four angles. They come in various shapes and sizes, each with its own unique properties. Here are some of the key properties of quadrilaterals:
 Sum of Interior Angles: The sum of the interior angles of any quadrilateral is always 360 degrees. This property holds true for all quadrilaterals, regardless of their shape or size.
Example: Consider a rectangle. The rectangle has four right angles, each measuring 90 degrees. Adding up the four angles, we get 90 + 90 + 90 + 90 = 360 degrees.
 Opposite Sides: In a quadrilateral, the opposite sides are always parallel and of equal length. This property is true for all quadrilaterals, except for parallelograms.
Example: Consider a square. The square has four equal sides, and the opposite sides are parallel to each other.
 Diagonals: The diagonals of a quadrilateral bisect each other. This property is true for all quadrilaterals, except for parallelograms.
Example: Consider a rhombus. The rhombus has two diagonals that intersect at right angles and bisect each other.

Types of Quadrilaterals: There are different types of quadrilaterals, each with its own specific properties. Some common types of quadrilaterals include:
 Rectangle: A rectangle is a quadrilateral with four right angles and opposite sides of equal length.
 Square: A square is a rectangle with all four sides of equal length.
 Rhombus: A rhombus is a quadrilateral with all four sides of equal length, but the angles are not necessarily right angles.
 Parallelogram: A parallelogram is a quadrilateral with opposite sides parallel and of equal length, but the angles are not necessarily right angles.
 Trapezoid: A trapezoid is a quadrilateral with at least one pair of parallel sides.
These are just a few of the properties of quadrilaterals. By understanding these properties, we can better understand and classify different types of quadrilaterals.
Sides and Angles of Quadrilaterals
Sides and Angles of Quadrilaterals
A quadrilateral is a polygon with four sides. The sum of the interior angles of a quadrilateral is 360 degrees.
Types of Quadrilaterals
There are many different types of quadrilaterals, but some of the most common include:
 Rectangle: A rectangle is a quadrilateral with four right angles.
 Square: A square is a rectangle with all four sides of equal length.
 Rhombus: A rhombus is a quadrilateral with all four sides of equal length, but not all angles are right angles.
 Parallelogram: A parallelogram is a quadrilateral with two pairs of parallel sides.
 Trapezoid: A trapezoid is a quadrilateral with only one pair of parallel sides.
Properties of Quadrilaterals
The properties of quadrilaterals can be used to identify different types of quadrilaterals. Some of the properties of quadrilaterals include:
 Sum of interior angles: The sum of the interior angles of a quadrilateral is 360 degrees.
 Opposite angles: The opposite angles of a quadrilateral are equal.
 Adjacent angles: The adjacent angles of a quadrilateral are supplementary.
 Diagonals: The diagonals of a quadrilateral bisect each other.
Examples of Quadrilaterals
Here are some examples of quadrilaterals:
 Rectangle: A rectangle is a quadrilateral with four right angles.
 Square: A square is a rectangle with all four sides of equal length.
 Rhombus: A rhombus is a quadrilateral with all four sides of equal length, but not all angles are right angles.
 Parallelogram: A parallelogram is a quadrilateral with two pairs of parallel sides.
 Trapezoid: A trapezoid is a quadrilateral with only one pair of parallel sides.
Applications of Quadrilaterals
Quadrilaterals are used in many different applications, such as:
 Architecture: Quadrilaterals are used in the design of buildings and other structures.
 Engineering: Quadrilaterals are used in the design of bridges, roads, and other infrastructure.
 Art: Quadrilaterals are used in the creation of paintings, drawings, and other works of art.
 Science: Quadrilaterals are used in the study of geometry, physics, and other scientific disciplines.
Quadrilaterals are a versatile and important geometric shape with many different applications.
Quadrilateral Formulas
Quadrilateral Formulas
In geometry, a quadrilateral is a polygon with four sides. There are many different types of quadrilaterals, but some of the most common include squares, rectangles, rhombuses, and trapezoids.
The formulas for finding the area and perimeter of a quadrilateral depend on the type of quadrilateral.
Square
A square is a quadrilateral with four equal sides and four right angles.
The area of a square is found by squaring the length of one side.
$$A = s^2$$
The perimeter of a square is found by multiplying the length of one side by 4.
$$P = 4s$$
Rectangle
A rectangle is a quadrilateral with four right angles, but not necessarily equal sides.
The area of a rectangle is found by multiplying the length by the width.
$$A = lw$$
The perimeter of a rectangle is found by adding the lengths of all four sides.
$$P = 2l + 2w$$
Rhombus
A rhombus is a quadrilateral with four equal sides, but not necessarily right angles.
The area of a rhombus is found by multiplying half the length of one diagonal by half the length of the other diagonal.
$$A = \frac{1}{2}d_1d_2$$
The perimeter of a rhombus is found by multiplying the length of one side by 4.
$$P = 4s$$
Trapezoid
A trapezoid is a quadrilateral with two parallel sides and two nonparallel sides.
The area of a trapezoid is found by multiplying the average of the lengths of the two parallel sides by the height.
$$A = \frac{1}{2}(b_1 + b_2)h$$
The perimeter of a trapezoid is found by adding the lengths of all four sides.
$$P = a + b_1 + b_2 + c$$
Examples
 A square with a side length of 5 cm has an area of 25 cm^2 and a perimeter of 20 cm.
 A rectangle with a length of 10 cm and a width of 5 cm has an area of 50 cm^2 and a perimeter of 30 cm.
 A rhombus with a side length of 6 cm and a diagonal length of 8 cm has an area of 24 cm^2 and a perimeter of 24 cm.
 A trapezoid with a base length of 8 cm, a top length of 4 cm, and a height of 5 cm has an area of 30 cm^2 and a perimeter of 20 cm.
Quadrilaterals Solved Examples
Quadrilaterals Solved Examples
Example 1: Find the area of a rectangle with length 10 cm and width 5 cm.
Solution:
The area of a rectangle is given by the formula:
Area = length × width
Substituting the given values, we get:
Area = 10 cm × 5 cm = 50 cm²
Therefore, the area of the rectangle is 50 cm².
Example 2: Find the perimeter of a square with side length 6 cm.
Solution:
The perimeter of a square is given by the formula:
Perimeter = 4 × side length
Substituting the given value, we get:
Perimeter = 4 × 6 cm = 24 cm
Therefore, the perimeter of the square is 24 cm.
Example 3: Find the diagonals of a rhombus with side length 8 cm and angle 60 degrees.
Solution:
The diagonals of a rhombus are given by the formulas:
d1 = 2 × side length × sin(angle/2)
d2 = 2 × side length × cos(angle/2)
Substituting the given values, we get:
d1 = 2 × 8 cm × sin(60°/2) = 8.94 cm
d2 = 2 × 8 cm × cos(60°/2) = 15.56 cm
Therefore, the diagonals of the rhombus are 8.94 cm and 15.56 cm.
Example 4: Find the area of a trapezoid with bases 8 cm and 12 cm, and height 5 cm.
Solution:
The area of a trapezoid is given by the formula:
Area = ½ × (sum of bases) × height
Substituting the given values, we get:
Area = ½ × (8 cm + 12 cm) × 5 cm = 50 cm²
Therefore, the area of the trapezoid is 50 cm².
Example 5: Find the perimeter of a parallelogram with sides 10 cm and 12 cm, and angle between them 60 degrees.
Solution:
The perimeter of a parallelogram is given by the formula:
Perimeter = 2 × (sum of sides)
Substituting the given values, we get:
Perimeter = 2 × (10 cm + 12 cm) = 44 cm
Therefore, the perimeter of the parallelogram is 44 cm.
Practice Questions on Quadrilaterals
Practice Questions on Quadrilaterals
1. A quadrilateral has four sides and four angles. What is the sum of the interior angles of a quadrilateral?
Answer: The sum of the interior angles of a quadrilateral is 360 degrees.
2. A parallelogram is a quadrilateral with opposite sides parallel and equal. What are the properties of a parallelogram?
Answer: The properties of a parallelogram are:
 Opposite sides are parallel and equal.
 Opposite angles are equal.
 Consecutive angles are supplementary.
 The diagonals bisect each other.
3. A rectangle is a parallelogram with four right angles. What are the properties of a rectangle?
Answer: The properties of a rectangle are:
 All sides are equal.
 All angles are right angles.
 The diagonals are equal and bisect each other.
4. A square is a rectangle with all sides equal. What are the properties of a square?
Answer: The properties of a square are:
 All sides are equal.
 All angles are right angles.
 The diagonals are equal and bisect each other.
 The square is a regular polygon.
5. A rhombus is a parallelogram with all sides equal. What are the properties of a rhombus?
Answer: The properties of a rhombus are:
 All sides are equal.
 Opposite angles are equal.
 The diagonals are perpendicular to each other.
 The rhombus is a regular polygon.
6. A trapezoid is a quadrilateral with one pair of parallel sides. What are the properties of a trapezoid?
Answer: The properties of a trapezoid are:
 One pair of sides are parallel.
 The opposite angles are equal.
 The diagonals bisect each other.
7. A kite is a quadrilateral with two pairs of adjacent sides equal. What are the properties of a kite?
Answer: The properties of a kite are:
 Two pairs of adjacent sides are equal.
 The opposite angles are equal.
 The diagonals are perpendicular to each other.
8. A deltoid is a quadrilateral with all sides equal. What are the properties of a deltoid?
Answer: The properties of a deltoid are:
 All sides are equal.
 The opposite angles are equal.
 The diagonals bisect each other.
 The deltoid is a regular polygon.
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Frequently Asked Questions
What are Quadrilaterals and their types?
Quadrilaterals
A quadrilateral is a polygon with four sides. Quadrilaterals can be classified into several types based on the lengths of their sides and the angles between them.
Types of Quadrilaterals
The following are some of the most common types of quadrilaterals:
 Parallelogram: A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length.
 Rectangle: A rectangle is a parallelogram with four right angles.
 Square: A square is a rectangle with all four sides equal in length.
 Rhombus: A rhombus is a parallelogram with all four sides equal in length. The angles of a rhombus are not necessarily right angles.
 Trapezoid: A trapezoid is a quadrilateral with only one pair of parallel sides.
 Kite: A kite is a quadrilateral with two pairs of adjacent sides that are equal in length. The opposite angles of a kite are equal.
Properties of Quadrilaterals
The following are some of the properties of quadrilaterals:
 The sum of the interior angles of a quadrilateral is 360 degrees.
 The opposite sides of a parallelogram are equal in length.
 The diagonals of a parallelogram bisect each other.
 The diagonals of a rectangle are perpendicular to each other.
 The diagonals of a square are perpendicular to each other and bisect each other.
 The diagonals of a rhombus are perpendicular to each other.
 The diagonals of a trapezoid bisect each other.
 The diagonals of a kite intersect at right angles.
Examples of Quadrilaterals
Here are some examples of quadrilaterals:
 A sheet of paper is a rectangle.
 A door is a rectangle.
 A window is a rectangle.
 A table top is a rectangle.
 A chair seat is a square.
 A diamond is a rhombus.
 A kite is a kite.
 A trapezoid is a trapezoid.
Quadrilaterals are used in a variety of applications, such as architecture, engineering, and design.
What is a convex and concave quadrilateral?
Convex Quadrilateral
A convex quadrilateral is a quadrilateral in which all interior angles are less than 180 degrees. In other words, no angle in a convex quadrilateral is greater than a straight angle.
Properties of Convex Quadrilaterals
 The sum of the interior angles of a convex quadrilateral is 360 degrees.
 The opposite sides of a convex quadrilateral are parallel.
 The diagonals of a convex quadrilateral intersect inside the quadrilateral.
Examples of Convex Quadrilaterals
 Square
 Rectangle
 Rhombus
 Parallelogram
 Trapezoid
Concave Quadrilateral
A concave quadrilateral is a quadrilateral in which at least one interior angle is greater than 180 degrees. In other words, at least one angle in a concave quadrilateral is greater than a straight angle.
Properties of Concave Quadrilaterals
 The sum of the interior angles of a concave quadrilateral is greater than 360 degrees.
 The opposite sides of a concave quadrilateral are not parallel.
 The diagonals of a concave quadrilateral intersect outside the quadrilateral.
Examples of Concave Quadrilaterals
 Kite
 Dart
 Crescent
 Boomerang
What is the sum of the interior angles of a quadrilateral?
The sum of the interior angles of a quadrilateral is 360 degrees. This can be proven using a variety of methods, one of which is by using the fact that the sum of the angles of a triangle is 180 degrees.
Consider a quadrilateral ABCD. We can draw a diagonal from vertex A to vertex C, dividing the quadrilateral into two triangles, ABC and ADC. The sum of the angles of triangle ABC is 180 degrees, and the sum of the angles of triangle ADC is also 180 degrees. Therefore, the sum of the angles of quadrilateral ABCD is 180 degrees + 180 degrees = 360 degrees.
This proof can be generalized to any quadrilateral, regardless of its shape or size. Therefore, the sum of the interior angles of a quadrilateral is always 360 degrees.
Here are some examples of quadrilaterals and the sums of their interior angles:
 A square has four right angles, so the sum of its interior angles is 4 * 90 degrees = 360 degrees.
 A rectangle has four right angles, so the sum of its interior angles is 4 * 90 degrees = 360 degrees.
 A rhombus has four equal angles, so the sum of its interior angles is 4 * (360 degrees / 4) = 360 degrees.
 A parallelogram has two pairs of equal angles, so the sum of its interior angles is 2 * (180 degrees / 2) + 2 * (180 degrees / 2) = 360 degrees.
The sum of the interior angles of a quadrilateral is a fundamental property of quadrilaterals, and it has many applications in geometry and other areas of mathematics.
What are the three attributes of a quadrilateral?
The three attributes of a quadrilateral are:

Opposite sides are parallel and equal in length. This means that the opposite sides of a quadrilateral are always the same length and are always parallel to each other. For example, in a rectangle, the opposite sides are the two long sides and the two short sides. The long sides are parallel to each other, and the short sides are parallel to each other.

Opposite angles are equal in measure. This means that the opposite angles of a quadrilateral are always the same size. For example, in a square, the opposite angles are all right angles.

The sum of the interior angles is 360 degrees. This means that the sum of the measures of all four interior angles of a quadrilateral is always 360 degrees. For example, in a rhombus, the sum of the interior angles is 360 degrees.
Here are some examples of quadrilaterals:
 Rectangle: A rectangle is a quadrilateral with four right angles. The opposite sides of a rectangle are parallel and equal in length.
 Square: A square is a quadrilateral with four equal sides and four right angles.
 Rhombus: A rhombus is a quadrilateral with four equal sides. The opposite angles of a rhombus are equal in measure, but the angles are not necessarily right angles.
 Parallelogram: A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, but the angles are not necessarily right angles.
 Trapezoid: A trapezoid is a quadrilateral with one pair of parallel sides. The opposite sides of a trapezoid are not necessarily equal in length, and the angles are not necessarily right angles.
How to find the perimeter of a quadrilateral?
Perimeter of a Quadrilateral
The perimeter of a quadrilateral is the sum of the lengths of all four sides. To find the perimeter of a quadrilateral, you need to know the lengths of all four sides.
Example 1: Rectangle
A rectangle is a quadrilateral with four right angles. The opposite sides of a rectangle are equal in length. To find the perimeter of a rectangle, you need to know the length and width of the rectangle.
The formula for the perimeter of a rectangle is:
P = 2L + 2W
where:
 P is the perimeter of the rectangle
 L is the length of the rectangle
 W is the width of the rectangle
For example, if a rectangle has a length of 10 cm and a width of 5 cm, the perimeter of the rectangle is:
P = 2(10 cm) + 2(5 cm)
P = 20 cm + 10 cm
P = 30 cm
Example 2: Square
A square is a quadrilateral with four equal sides. To find the perimeter of a square, you only need to know the length of one side.
The formula for the perimeter of a square is:
P = 4S
where:
 P is the perimeter of the square
 S is the length of one side of the square
For example, if a square has a side length of 5 cm, the perimeter of the square is:
P = 4(5 cm)
P = 20 cm
Example 3: Parallelogram
A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length. To find the perimeter of a parallelogram, you need to know the length of one side and the length of one of the parallel sides.
The formula for the perimeter of a parallelogram is:
P = 2(L + S)
where:
 P is the perimeter of the parallelogram
 L is the length of one side of the parallelogram
 S is the length of one of the parallel sides of the parallelogram
For example, if a parallelogram has a side length of 10 cm and a parallel side length of 5 cm, the perimeter of the parallelogram is:
P = 2(10 cm + 5 cm)
P = 2(15 cm)
P = 30 cm
Example 4: Trapezoid
A trapezoid is a quadrilateral with one pair of parallel sides. The opposite sides of a trapezoid are not equal in length. To find the perimeter of a trapezoid, you need to know the lengths of all four sides.
The formula for the perimeter of a trapezoid is:
P = A + B + C + D
where:
 P is the perimeter of the trapezoid
 A is the length of one side of the trapezoid
 B is the length of the opposite side of the trapezoid
 C is the length of one of the parallel sides of the trapezoid
 D is the length of the other parallel side of the trapezoid
For example, if a trapezoid has side lengths of 10 cm, 15 cm, 8 cm, and 12 cm, the perimeter of the trapezoid is:
P = 10 cm + 15 cm + 8 cm + 12 cm
P = 45 cm