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Quadrilateral – Definition, Properties, Types, FAQs, Examples

Quadrilaterals are fascinating. From a simple rectangle to a complex kite, these geometric shapes have different properties and characteristics that make them fun and exciting to study. Read on to learn about the world of quadrilateral shapes, their properties, classification, and applications in real-world problems. Whether you’re a math teacher or a parent who wants to teach your child about geometry, this article will offer insights and inspiration for the quadrilateral concept.

What’s a Quadrilateral?

The quadrilateral definition is quite simple. So, what is quadrilateral? It is a four-sided polygon, which means it is a geometric shape with four straight sides and four angles. It is derived from the Latin words “Quadri,” meaning “four,” and “latus,” meaning “side.” The study of quadrilaterals falls under the branch of mathematics known as geometry and has applications in fields such as architecture and engineering. 

Types of Quadrilaterals

There are many different types of quadrilaterals, some of which you already know. Each one has its own rules and properties. Here are some of them: 

Square: A square is a quadrilateral with four equal sides and four right angles.

Rectangle: A rectangle is a quadrilateral with four right angles, but only two parallel sides are equal in length.

Rhombus: A rhombus quadrilateral has four equal sides, but only opposite angles are equal.

Parallelogram: A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length.

Trapezoid: A trapezoid is a quadrilateral with a pair of parallel sides.

Kite: A kite is a quadrilateral shape with two pairs of adjacent sides of equal lengths. 

Convex, Concave, and Intersecting Quadrilaterals

You can classify quadrilaterals as convex, concave, or intersecting quadrilaterals based on shape and orientation. Here is how they are classified accordingly: 

Convex Quadrilateral

A convex quadrilateral is a polygon that has four sides and four angles. If all the angles inside the quadrilateral are less than 180 degrees and none of the sides is bent inward or “caved in,” the quadrilateral is considered convex. Convex quadrilaterals have outward-pointing vertices, and their opposite sides are parallel. Additionally, a convex quadrilateral has two diagonals that intersect inside the shape, dividing each other into two equal parts.

Concave Quadrilateral

A concave quadrilateral is a polygon with four sides and four interior angles, but at least one of its angles measures more than 180 degrees, which means that at least one of its sides is bent inward or “caving in.” In a concave quadrilateral, at least one of the vertices points inward. This inward-pointing vertex causes the polygon to have a hollow or “cave-like” shape. Concave quadrilaterals can have properties that differ significantly from convex quadrilaterals, such as non-parallel opposite sides and diagonals that do not always intersect inside the shape.

Intersecting Quadrilateral

An intersecting quadrilateral is a polygon with four sides and four angles, but at least one pair of its sides cross each other instead of being parallel. When two sides of the quadrilateral cross, they form an intersection point where the sides meet. 

Intersecting quadrilaterals can have unique and complex properties compared to convex and concave quadrilaterals, such as unequal opposite sides, diagonals that do not intersect inside the shape, and different angles formed by the intersection of sides.

Properties of a Quadrilateral 

Quadrilateral properties are different. And often, the properties depend on the type of quadrilaterals. However, here are some of the common properties of all the quadrilaterals:

• All quadrilaterals have four sides. These sides can either have equal or unequal lengths.
• All quadrilaterals have four angles. The sum of the interior angles of any quadrilateral is always 360 degrees.
• All quadrilaterals have two diagonals.
• A quadrilateral can be concave or convex. A convex quadrilateral has all its interior angles less than 180 degrees, and a concave quadrilateral has at least one interior angle greater than 180 degrees.
• In some quadrilaterals, such as parallelograms, opposite sides are parallel.
• In some quadrilaterals, such as kites and rhombi, opposite sides are congruent.
• In some quadrilaterals, such as rectangles and squares, opposite angles are congruent.
• The sum of the opposite angles is always 180 degrees in quadrilaterals such as trapezoids.

Sides and Angles of a Quadrilateral 

The angles and sides of a quadrilateral depend on its type. In general, to find the angle of a quadrilateral, you can use the formula:

(angle sum) = 360 degrees

For example, in a parallelogram, opposite angles are congruent, so you can use this property to find the measure of an angle if you know the measure of the opposite angle. All four angles are right angles in a rectangle, so each angle measures 90 degrees. All four angles are also correct in a square, so each angle measures 90 degrees. In a rhombus, opposite angles are congruent but not right angles. In a trapezoid, the measure of the angles depends on the shape of the trapezoid.

Quadrilateral Formulas 

Here are the formulas used to calculate a quadrilateral shape: 

Perimeter: The perimeter of a quadrilateral is the sum of the lengths of its four sides. If the sides have lengths a, b, c, and d, then the perimeter P is:

P = a + b + c + d

Area: The formula for a quadrilateral’s area often depends on its type. The formula used to calculate the area of a quadrilateral is base multiplied by height, which is usually written as b*h. The unit of measurement for the area is square meters (m^2). But when studying different quadrilaterals, you will use the formula attached to that quadrilateral. 

Solved Examples of Quadrilateral Shapes 

Here are some solved examples of quadrilateral:

Example 1

Find the area of a rectangle if its length is 8 cm and width is 5 cm.

Solution:

The formula A = length x width gives a rectangle’s area.

Substituting the given values, we get:

A = 8 cm x 5 cm = 40 cm^2

Therefore, the area of the rectangle is 40 square centimeters.

Example 2

Find the perimeter of a square if its side length is 12 cm.

Solution:

The perimeter of a square is the sum of the lengths of its four sides.

Since all sides of a square are congruent, we can use the formula:

P = 4s, where s is the length of one side.

Substituting s = 12 cm, we get:

P = 4 x 12 cm = 48 cm

Therefore, the perimeter of the square is 48 cm.

Example 3

Find the length of the diagonal of a rhombus if its sides have a length of 5 cm and one of its angles measures 60 degrees.

Solution:

In a rhombus, the diagonals bisect each other at right angles, dividing the rhombus into four congruent right triangles. For example, the diagonal that bisects the 60-degree angle divides the rhombus into two 30-60-90 right triangles.

Using the properties of 30-60-90 triangles, we know that the hypotenuse (i.e., the diagonal of the rhombus) is twice the length of the shorter leg. Since the shorter leg of the right triangle is half the length of one side of the rhombus, we can find the length of the diagonal as follows:

Shorter leg = (1/2) x 5 cm = 2.5 cm

Diagonal = 2 x shorter leg = 2 x 2.5 cm = 5 cm

Therefore, the length of the diagonal of the rhombus is 5 cm.

Practice Questions on Quadrilaterals 

• Find the perimeter of a rectangle if its length is 12 meters and width is 5 meters.
• Find the area of a square if its side length is 9 cm.
• A parallelogram has a base of 10 meters and a height of 6 meters. Find its area.
• Find the length of the diagonal of a square if its side length is 10 cm.
• A trapezoid has a height of 12 cm, and its parallel sides are 8 cm and 12 cm. Find its area.
• Find the length of the diagonal of a rhombus if its sides have a length of 7 cm and one of its angles measures 120 degrees.
• A quadrilateral has sides of 6 cm, 8 cm, 10 cm, and 12 cm. Can this quadrilateral be a square? Why or why not?
• Find the area of a kite if its diagonals have lengths of 6 cm and 8 cm.
• Find the perimeter of a rhombus if its diagonals have lengths of 10 cm and 12 cm.
• A quadrilateral has angles of 90 degrees, 100 degrees, 80 degrees, and 110 degrees. Can this quadrilateral be a parallelogram? Why or why not?

Conclusion 

Learning about quadrilaterals can be challenging because of all the many moving parts. But this guide can help you understand quadrilaterals in more detail, especially when you are practicing for an exam or a standardized test.

Frequently Asked Questions on Quadrilateral

What are the different types of quadrilaterals?

The quadrilaterals are square, rectangle, rhombus, parallelogram, and trapezoid.

What is the sum of the interior angles in a quadrilateral?

The sum of the interior of the angles in a quadrilateral is 360 degrees. 

What are the common properties of all quadrilaterals?

They all have four sides and four angles. 

How to find the area of a quadrilateral?

The general formula for the area of quadrilateral is base*height with the unit of the measurement being m2.

How to find the perimeter of a quadrilateral?

P = a + b + c + d

How many sides does a quadrilateral have?

A quadrilateral has four sides.