Special quadrilaterals

Have you ever looked at a chess or checkers board and thought to yourself,  "Wow, this is a fine-looking perfect square!"? Without realizing it, this square chess board is considered a special quadrilateral. In this article, we will discuss:

Special quadrilaterals Special quadrilaterals

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Contents
Table of contents
    • Five types of special quadrilaterals (kites, rhombuses, rectangles, squares, and trapezoids)

    • Some properties of each

    For more information on each type of these special quadrilaterals, check out our articles on Rectangles, Squares, Rhombuses, and Trapezoids.

    First, let's review quadrilaterals. You can find more details in Quadrilaterals.

    A Review of Quadrilaterals

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

    A quadrilateral is a two-dimensional figure. It is a type of polygon, and what makes a figure a quadrilateral is the fact that it has four sides.

    "Quadri" in the word "quadrilateral" is a Latin word that means 4. The word "lateral" means "sides. That is why a figure with four sides is called a quadrilateral.

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

    Special Quadrilaterals shape with unequal sides and angles StudySmarter A shape with no equal sides or angles - StudySmarter Originals

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

    Special Quadrilaterals Rectangle StudySmarterA 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?

    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). Refer to the article on Polygons for specifics!

    The five types of special quadrilaterals are:

    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.

    Special Quadrilaterals Rectangle StudySmarterA 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.

    Remember that to bisect means to divide into parts.

    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.

    Special Quadrilaterals Square StudySmarterA 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.

    Special Quadrilaterals Rhombus StudySmarterA 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.

    A rhombus, square, and rectangle are known as parallelograms. You can find out more about them from our article on the Properties of parallelograms.

    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.

    Special Quadrilaterals Trapezoid StudySmarterA 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.

    Special Quadrilaterals  kite StudySmarterA 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:

    Special quadrilaterals Rectangle StudySmarterA diagram of a rectangle - StudySmarter Originals

    Area = L × B

    Perimeter = L + L + B + BPerimeter = 2 × L + 2 × BPerimeter = 2 × (L + B)

    Square

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

    Special quadrilaterals Square StudySmarterA diagram of a square - StudySmarter Originals

    Area = L × L = L2

    Perimeter = 2 × (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.

    Special Quadrilaterals Rhombus StudySmarterA diagram of a rhombus - StudySmarter Originals

    Area = 12 × d1 × d2

    Perimeter = x + x + x + xPerimeter = 4x

    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.

    Special Quadrilaterals Trapezoid StudySmarterA diagram of a trapezoid - StudySmarter Originals

    Area = 12 × (AB + CD) × h

    Perimeter = AB + CD + BC + DA

    Kite

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

    Special Quadrilaterals Kite StudySmarterA diagram of a kite, StudySmarter Originals

    In the diagram above, the area is

    Area = BC¯×AD¯2

    Special Quadrilaterals - Key takeaways

    • Special quadrilaterals are quadrilaterals with some or all equal sides and angles.
    • Special quadrilaterals are kites, squares, rhombuses, rectangles, and trapezoids.
    • Special quadrilaterals have individual properties as well as some common ones.
    • The formula used to calculate the area and perimeter of special quadrilaterals depends on the quadrilateral in question.
    Frequently Asked Questions about Special quadrilaterals

    What are special quadrilaterals?

    Special quadrilaterals are quadrilaterals with some or all equal sides and angles.

    What makes a quadrilateral?

    A quadrilateral is made up of four sides and four angles.

    What are the properties of special quadrilaterals?

    The properties of special quadrilaterals depends on the type of quadrilateral that is being examined.

    What are examples of special quadrilaterals?

    Examples of special quadrilaterals are square, rectangle, rhombus and trapezoid.

    What is the formula used for special quadrilaterals?

    The formula used for special quadrilaterals depends on the type of quadrilateral in questions. There are different formulas used to calculate the area and perimeter of different special quadrilaterals.

    Test your knowledge with multiple choice flashcards

    A square is a special case of a parallelogram. Is this true or false?

    A square has all the properties of a rectangle, but not the ones from a rhombus. Is this true or false?

    A square, a rectangle and a rhombus are all special cases of a quadrilateral, except for the parallelogram. Is this true or false?

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