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Special quadrilaterals

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:

  • 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

hh

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

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

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

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

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

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

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

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

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

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.

Final Special quadrilaterals Quiz

Question

Is a rectangle a special case of parallelogram?

Show answer

Answer

Yes

Show question

Question

Is a square a special case of a rectangle?

Show answer

Answer

Yes

Show question

Question

Can the diagonal of a rectangle be perpendicular? 

Show answer

Answer

No

Show question

Question

In a rectangle the opposite sides are equal and parallel. Is is true or false?

Show answer

Answer

True

Show question

Question

In a rectangle at least 2 angles must be right angles. Is it true or false?

Show answer

Answer

False

Show question

Question

The perimeter of a flat shape or the surface of an object can be defined in geometry as the space occupied by it. Is is true or False?


Show answer

Answer

False

Show question

Question

All rectangles are squares. Is it true or false?

Show answer

Answer

False

Show question

Question

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

Show answer

Answer

True

Show question

Question

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

Show answer

Answer

False

Show question

Question

A square, a rectangle and a rhombus have all the properties of a parallelogram. Is this true or false? 

Show answer

Answer

True 

Show question

Question

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


Show answer

Answer

False

Show question

Question

The opposite sides of the square are parallel to each other. Is this true or false?

Show answer

Answer

True

Show question

Question

The diagonals of the square bisect each other at 90°. Is this true or false?


Show answer

Answer

True

Show question

Question

The perimeter of a square is equal to its side's length, raised to the power of 2. Is this true or false?

Show answer

Answer

False

Show question

Question

The area of a square is equal to its side's length multiplied by four. Is this true or false?

Show answer

Answer

False

Show question

Question

Determine the area of a square with a side length of 6 m.

Show answer

Answer

36

Show question

Question

If a square has its sides equal to 5 cm each. What will be its perimeter? 


Show answer

Answer

20 cm

Show question

Question

If the perimeter of a square is 36 m, then what is the side of the square?  


Show answer

Answer

9

Show question

Question

Find the area of a square is its perimeter is equal to 36 m.  


Show answer

Answer

81 m2

Show question

Question

The square is a 2D shape with four equal sides and four 90° angles. Is this true or false?


Show answer

Answer

True

Show question

Question

What are special quadrilaterals?

Show answer

Answer

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

Show question

Question

Which one of the following is NOT an example of special quadrilaterals ?

A. Rectangle

B. Rhombus

C. Circle

D. Square

Show answer

Answer

C

Show question

Question

List some properties of a rectangle.

Show answer

Answer

  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.

Show question

Question

List some properties of a square.

Show answer

Answer

  1. All sides of a square are equal and parallel to each other.
  2. It has four equal angles of 90 degrees.
  3. The diagonals bisect each angle.
  4. The opposite sides are parallel.
  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.

Show question

Question

List some properties of a rhombus.

Show answer

Answer

  1. The opposite sides of a rhombus are equal and parallel.
  2. The angles of a rhombus are equal.
  3. The diagonals bisect each other.
  4. The diagonals of a rhombus bisect each angle.

Show question

Question

List some properties of a trapezoid.

Show answer

Answer

  1. Just one pair of the opposite sides (the base) are parallel to each other.
  2. The sum of the angles next to each other is 180 degrees.
  3. The non-parallel sides of a trapezoid are equal.

Show question

Question

What is a quadrilateral?

Show answer

Answer

A quadrilateral is a type of polygon that has four sides. 

Show question

Question

Give some examples of special quadrilateral.

Show answer

Answer

  • Rectangle
  • Rhombus
  • Square
  • Trapezoid

Show question

Question

What are some common properties between special quadrilaterals?

Show answer

Answer

1.  A rectangle and a square have four equal angles of 90 degrees each.

2.  The diagonals intersects each other and bisects each angle.

3.  The opposite sides are parallel to each other. 

4.  A square, rectangle rhombus and trapezoid are all parallelograms. 

Show question

Question

What are the properties a quadrilateral must posses before it can be called a parallelogram? 

Show answer

Answer

  1. Both pairs of opposite sides are equal.
  2. One pair of the opposite side is equal.
  3. Both pairs of opposite angles are equal.
  4. The diagonals bisect each other.

Show question

Question

What is a rhombus?

Show answer

Answer

A rhombus is a quadrilateral whose all 4 sides are equal.

Show question

Question

State whether the following statement is true or false:

Every parallelogram is a rhombus.

Show answer

Answer

False

Show question

Question

State whether the following statement is true or false:

Every rhombus is a parallelogram.

Show answer

Answer

True

Show question

Question

State whether the following statement is true or false:

The opposite angles of a rhombus are supplementary.

Show answer

Answer

False

Show question

Question

State whether the following statement is true or false:

The diagonals of a rhombus bisect each other.

Show answer

Answer

True

Show question

Question

A rhombus with an area of 96 has a diagonal of length 12. Find the length of the other diagonal.

Show answer

Answer

8

Show question

Question

A rhombus with an area of 18 has a diagonal of length 6. Find the length of the other diagonal.

Show answer

Answer

6

Show question

Question

State whether the following statement is true or false:

Every parallelogram is a rhombus.

Show answer

Answer

false

Show question

Question

State whether the following statement is true or false:

Every trapezium is a rhombus.

Show answer

Answer

false

Show question

Question

State whether the following statement is true or false:

Opposite angles of a rhombus are equal.

Show answer

Answer

True

Show question

Question

State whether the following statement is true or false: 

The diagonals of a rhombus are perpendicular to each other.

Show answer

Answer

True


Show question

Question

State whether the following statement is true or false: 

If the diagonals of a parallelogram are perpendicular, then it implies that the given parallelogram is a rhombus.

Show answer

Answer

True

Show question

Question

A parallelogram ABCD has diagonals AC and BD such that AC ⊥ BD. Can we conclude that ABCD is a rhombus?

Show answer

Answer

Yes

Show question

Question

State whether the following statement is true or false: Not every square is a rhombus.

Show answer

Answer

False

Show question

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