# Special quadrilaterals

## Demystifying Quadrilaterals: Defining and Understanding Special Types

Have you ever stopped to admire the intricacies of a chess or checkers game board and marveled at its perfectly symmetrical, four-sided shape? Did you know that this shape is actually a unique type of quadrilateral? In this article, we will explore:

- Five special types of quadrilaterals - kites, rhombuses, rectangles, squares, and trapezoids
- The distinct properties of each type

For more in-depth information on these special quadrilaterals, be sure to check out our articles on Rectangles, Squares, Rhombuses, and Trapezoids.

## A Comprehensive Look at Quadrilaterals

Before we delve into the specifics of special quadrilaterals, let's first review the definition of a "quadrilateral". For a more detailed explanation, see our article on Quadrilaterals.

A quadrilateral is a two-dimensional shape that falls under the category of a polygon. What sets quadrilaterals apart is the fact that they have four sides.

The word "quadrilateral" comes from the Latin words "quadri", meaning "four", and "lateral", meaning "sides". This is why a shape with four sides is referred to as a quadrilateral.

Some quadrilaterals have no equal sides or angles, as shown in the figure below.

**A shape with no equal sides or angles**

Others have pairs of congruent sides and angles, like a rectangle shown in the diagram below.

**A diagram of a rectangle**

For the purpose of this article, we will focus on the latter group - quadrilaterals that have some combination of equal side lengths and/or angles.

## What Sets Special Quadrilaterals Apart?

Special quadrilaterals are polygons with four sides that have some combination of equal side lengths and/or angles.

A polygon is a two-dimensional enclosed figure, such as a triangle or square. For a more detailed explanation, refer to our article on Polygons.

The five types of special quadrilaterals are:

- Rectangle
- Rhombus
- Square
- Trapezoid
- Kite

Now, let's take a closer look at the unique properties of each of these special quadrilaterals.

## The Properties of Special Quadrilaterals

As we explore the properties of special quadrilaterals, you will notice that some have similarities, but each also has its own distinct characteristics. Let's begin with the rectangle.

### Rectangle:

A rectangle is a quadrilateral with opposite parallel sides of equal length and four equal right angles of 90 degrees each. The figure below shows a rectangle.

**Properties of a Rectangle:**

- Opposite sides are equal and parallel
- Has four angles of 90 degrees each
- Diagonals bisect each other
- The sum of angles in a rectangle is 360 degrees

### Square:

A square also has opposite sides parallel to each other and four equal angles and sides. Like a rectangle, the angles are all 90 degrees. The diagram below shows a square for reference.

**Properties of a Square:**

- All sides are equal
- Opposite sides are parallel
- Diagonals bisect each angle
- Has four angles of 90 degrees each
- The length of diagonals is longer than the sides of the square
- The sum of angles in a square is 360 degrees

### Rhombus:

A rhombus is a quadrilateral with equal sides, and the opposite sides are parallel to each other. It also has equal opposite angles. Refer to the diagram below for a visual representation of a rhombus.

**Properties of a Rhombus:**

- Opposite sides are equal and parallel
- Opposite angles are equal
- Diagonals bisect each other
- Diagonals bisect each angle

A rhombus, square, and rectangle are all considered parallelograms. For more information on parallelograms, see our article on the properties of parallelograms.

### Trapezoid:

A trapezoid is a quadrilateral that is not classified as a parallelogram because it has one pair of parallel sides. These parallel sides are known as bases, and the other two sides are referred to as legs. The angles in a trapezoid are called base angles.

**Understanding Special Quadrilaterals: Area and Perimeter Formulas**

- Special quadrilaterals have certain properties that make them unique and different from other quadrilaterals.
- The formula for finding their area and perimeter varies depending on the type of special quadrilateral.

To learn more about special quadrilaterals and their individual properties, continue reading below.

### Rectangle:

A rectangle is a quadrilateral with opposite sides that are equal in length, making all four angles 90 degrees. It is also known as a parallelogram with four right angles. Its area can be calculated by multiplying the length and width, and its perimeter is found by adding all four sides.

**Properties of a Rectangle:**

- All four sides are equal in length
- All four angles are 90 degrees

### Rhombus:

A rhombus is a quadrilateral with four equal sides, making all four angles equal to each other. Its area is half the product of its diagonals, and its perimeter is four times the length of each side.

**Properties of a Rhombus:**

- All four sides are equal in length
- All four angles are equal
- Diagonals bisect each other at right angles

### Trapezoid:

A trapezoid is a quadrilateral with one pair of parallel sides. Its area can be found by taking half the product of the sum of the parallel sides and the height. The perimeter is calculated by adding all four sides together.

**Properties of a Trapezoid:**

- One pair of opposite sides (the bases) are parallel
- Sum of any two adjacent angles is 180 degrees

### Kite:

The final type of special quadrilateral is the kite. It has two pairs of congruent sides that are adjacent to each other. The area of a kite is half the product of its diagonals, while its perimeter is the sum of all four sides. See diagrams below for visual representation and formulas.

**Properties of a Kite:**

- Two pairs of congruent sides
- Two pairs of adjacent sides
- Angles opposite each other from the diagonals are equal

**Example diagram of a kite**

In the diagram above, the area would be calculated as half the product of the length of the diagonals.

### Key Takeaways - Special Quadrilaterals

- Special quadrilaterals have unique properties such as equal sides and angles.
- Their area and perimeter formulas vary depending on the type of quadrilateral.
- Examples of special quadrilaterals include squares, rectangles, rhombuses, and trapezoids.
- These quadrilaterals have individual properties as well as some common characteristics.

**What are Special Quadrilaterals?**

Special quadrilaterals are quadrilaterals with specific properties that make them different from other quadrilaterals.

**What Makes a Quadrilateral?**

A quadrilateral is a shape with four sides and four angles.

**What are the Properties of Special Quadrilaterals?**

The properties of special quadrilaterals vary depending on the type of quadrilateral being studied.

**What are Examples of Special Quadrilaterals?**

Some examples of special quadrilaterals are squares, rectangles, rhombuses, and trapezoids.

**What is the Formula Used for Special Quadrilaterals?**

The formula used to find the area and perimeter of special quadrilaterals depends on the specific type of quadrilateral being examined. Different formulas are used to calculate the area and perimeter of each special quadrilateral.