Special quadrilaterals

A Review of Quadrilaterals

Let's recall what we mean when we say "quadrilateral". For more details, see our article on Quadrilaterals.

Some quadrilaterals don't have any matching side lengths or angles, like in the figure below.

A shape with no equal sides or angles - StudySmarter Originals

Some have pairs of congruent side lengths and angles, like in the figure below:

A diagram of a rectangle - StudySmarter Originals

This article will focus on the latter of these two examples (quadrilaterals that have some combination of equal side lengths and/or angles). Let's have a further look!

What are Special Quadrilaterals?

The five types of special quadrilaterals are:

• Rectangle

• Rhombus

• Square

• Trapezoid

• Kites

Let's take a look at the properties of each quadrilateral.

Types of Special Quadrilaterals and their Properties

Now, let's look at what makes the special quadrilaterals so special. They each have their unique properties, but you will find that some of them have some similarities. Let's start with a 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.

A diagram of a rectangle - StudySmarter Originals

Properties of a Rectangle

1. The opposite sides of a rectangle are equal and parallel to each other.

2. It has four equal angles of 90 degrees.

3. The diagonals bisect each other.

4. The sum of the angles in a rectangle is 360 degrees.

Square

A square is a quadrilateral with opposite sides parallel to each other. They also have four equal angles and sides. Just like the rectangle, the angles are 90 degrees each. The figure below is a square.

A diagram of a square - StudySmarter Originals

Properties of a square

1. All sides of a square are equal.

2. The opposite sides are parallel.

3. The diagonals bisect each angle.

4. It has four equal angles of 90 degrees.

5. The length of the diagonals is longer than the sides of the square.

6. The sum of the angles in a square is 360 degrees.

Rhombus

The rhombus is a quadrilateral with equal sides. The opposite sides are parallel to each other. A rhombus also has equal opposite angles.

A diagram of a rhombus - StudySmarter Originals

Properties of a Rhombus

1. The opposite sides of a rhombus are equal and parallel.

2. The opposite angles of a rhombus are equal.

3. The diagonals bisect each other.

4. The diagonals of a rhombus bisect each angle.

Trapezoid

A trapezoid is a quadrilateral. However, it is not a parallelogram because it has just one pair of parallel sides. The parallel sides are called the bases, and the other two sides are called the legs. The angles in a trapezoid are called base angles.

A diagram of a trapezoid - StudySmarter Originals

Properties of a Trapezoid

1. Just one pair of the opposite sides (the base) are parallel.

2. The sum of any 2 angles next to each other is 180 degrees.

There are a few types of trapezoids that give themselves additional unique properties. You can read more about these in the article on Trapezoids.

Kites

The kite is our final type of special quadrilateral. It has two pairs of congruent sides that are adjacent to each other. The figure below shows a kite.

A diagram of a kite - StudySmarter Originals

Properties of a kite

1. It has two pairs of congruent sides.
2. It has two pairs of adjacent sides.
3. The angles opposite each other from the diagonals are equal.
4. The diagonals bisect each other at right angles.

Special Quadrilateral Area and Perimeter Formulas

You may need to find the area or the perimeter of the quadrilaterals. The formula used to get these depends on the quadrilateral in question. You can find more details in the Area of plane figures folder. Let's see the various formulas used to get the area and perimeter of special quadrilaterals.

Rectangle

The area of a rectangle is the product of its length (L) and breadth (B). A rectangle has equal opposite sides. So, when finding the area, you need to use the value of one of its opposite sides. You can determine the perimeter by adding all the sides together. Let's look at the formulas:

A diagram of a rectangle - StudySmarter Originals

$Area=L\times B$

Square

A square has four equal sides. Let's look at the area and perimeter of a square.

A diagram of a square - StudySmarter Originals

$Area=L\times L=L^2$

$Perimeter=2\times (L+L)=4L$

Rhombus

The area of a rhombus is half the product of the length of its diagonals. The perimeter of a rhombus is the summation of all its equal sides or 4 times the side length.

A diagram of a rhombus - StudySmarter Originals

$$\begin{gathered} Perimeter=x+x+x+x \\ Perimeter=4x \end{gathered}$$

Trapezoid

The area of a trapezoid is half the product of the sum of the parallel sides and the height. The perimeter of a trapezoid is the sum of all the sides.

A diagram of a trapezoid - StudySmarter Originals

$Area=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\times (AB+CD)\times h$

$Perimeter=AB+CD+BC+DA$

Kite

The area is a kite if half the product of the diagonals.

A diagram of a kite, StudySmarter Originals

In the diagram above, the area is

$Area=\frac{\overline{BC}\times \overline{AD}}{2}$