This article is about the square as a geometric figure. We explain to you what properties a square has and what different types of squares there are.

## quadrilateral definition

A quadrilateral is a geometric figure formed by **four** pages and **four** corner points is formed.

The **vertices **usually denoted by the capital letters A, B, C, D in alphabetical order. However, the designation is counter-clockwise, as you can see from the following figure.

Figure 1: Representation of a square

The four **sides of the square **are formed between the nearest vertices and also **versus** clockwise with the lower case letters a, b, c, d. Thus, the side between points A and B is also referred to as a.

In addition to the sides that enclose the quadrilateral, there are lines within the quadrilateral that run between the individual points. These are also called **diagonals **defined and marked with the letters e and f. The diagonal e runs between the corner points A and C and the diagonal f between the points B and D.

Regardless of how the particular quadrilateral is constructed, each quadrilateral possesses **four **Angles that are always formed by two sides and a corner point. The angles of these vertices are given the corresponding lowercase letter of the Greek alphabet. This means that, for example, the angle that lies at corner point A is denoted by α (alpha, beta, gamma, delta). The sum of all angles within the quadrilateral is always **360 degrees.**

## squares types

Quadrilaterals are classified according to their various peculiarities in **general** squares and **specific **squares divided. In the following, we would like to briefly introduce you to the different squares and show you how to use them **Scope **(U) and the **surface area** (A) can calculate. This is not only of enormous importance at school, but you can use these formulas again and again in everyday situations – whether it is decorating your room or mowing the lawn.

### General square

If a quadrilateral does not have any special features apart from the four corners and four sides, it is also called a **general square**** **designated.

This means, for example, that the sides are not the same length. A general quadrilateral can have one of three shapes:

**Convex quadrilateral****Concave square****Overturned square**

Let’s take a closer look at the different square types!

#### Convex quadrilateral

We consider the convex quadrilateral as the first type of quadrilateral. The following definition applies to this quadrilateral!

A convex quadrilateral is a quadrilateral in which the diagonals **inside** of the square cut.

As you can see from the figure below, the intersection of the diagonals e and f is inside the quadrilateral.

Figure 2: Convex quadrilateral

#### Concave square

The concave quadrilateral differs significantly from the convex quadrilateral! In general, you can remember the following!

In contrast to the convex square, the two diagonals of the concave square intersect **outside of** of the square. Thus, one of the four corners is curved inwards.

If one lengthens the diagonal f, which runs between the corner points B and D, then this intersects the diagonal e outside the quadrilateral area.

Figure 3: Concave square

#### Overturned square

The flipped square looks very different compared to the previous two square types!

Under a** overturned** A square is a geometric figure in which the order of the corner points is changed and they are no longer next to each other. Consequently, the individual pages cross each other.

Figure 4: Overturned square

### Special squares

In addition to the general quadrilateral, there are also a large number of quadrilaterals that can be distinguished from one another on the basis of certain properties.

#### rectangle

You are probably most familiar with the rectangle as a special quadrilateral.

A rectangle, like any other quadrilateral, has **four** vertices and** four** Pages. However, all four interior angles each form a right angle, i.e. they have 90 degrees. As a result, the opposite sides are always the same length, as are the diagonals. A rectangle has two sides of equal length!

Figure 5: rectangle

The calculation of the perimeter and the area are of enormous importance.

**1. Scope**

The perimeter U is the length of the line that delimits an area. For a quadrilateral, the perimeter is the total of the side lengths.

You can also think of the circumference as the length of rope it would take you to go around the rectangle once. You can see the scope in the following figure on the orange marked sides.

Figure 6: Perimeter of a rectangle

You can use the following formula to calculate the perimeter of a rectangle:

**2. Area **

The area is the dimension of a flat, i.e. two-dimensional, figure. For a rectangle, you can easily calculate this by multiplying the length by the width.

Figure 7: Area of a rectangle

**Task:**

You want to renovate your room and would like to replace the old carpet with a PVC covering. Your room is 4 m long and 5.5 m wide. How many m² of PVC flooring do you need?

**Solution:**

To know how much PVC flooring you need, use the formula for the area of a rectangle:

a = 4 m

b = 5.5 m

You need 22 m² of PVC flooring to replace the old carpet in your room.

#### The square as a special form of the rectangle

The square, as a special form of the rectangle, has four **same** long sides. All interior angles of a square are 90 degrees.

Since all sides of a square are the same length, the diagonals intersect at right angles.

Figure 8: square

So that you can also calculate the perimeter (U) and the area (A) of a square, we provide you with the following formulas:

#### parallelogram

A parallelogram is another special quadrilateral. In a parallelogram, the opposite sides of the quadrilateral are always **parallel** to each other. For this reason, the opposite angles are always the same size.

Of course you can also calculate the perimeter (U) and the area (A) of a parallelogram. However, in addition to the sides a and b, you also need the height h.

**A = a + b + c + d = 2 ⋅ a + 2 ⋅ b = 2 ⋅ (a + b) **

**A = a ⋅ h**

#### Rhombus

The rhombus is also known as a rhombus and is also a special square. The sides are not only parallel, but also of equal length. The angles of the rhombus are bisected by the diagonals.

The following formulas will help you to calculate the perimeter (U) and the area (A) of a rhombus:

#### trapeze

A trapezoid is another special quadrilateral. The special feature of the trapeze is that at least two opposite sides are parallel to each other. The parallel sides are called a trapezoid** base pages**the other two pages are the** leg**. In a trapezoid, the sum of the angles on one side is always 180 degrees.

Figure 11: trapeze

You will also be able to calculate the perimeter (U) and the area (A) of the trapezoid using the following formulas:

**A = a + b + c + d**

#### kite square

A kite is a quadrilateral with one diagonal as a **axis of symmetry **of the square acts. In a kite, two adjacent sides are always the same length. The diagonals are also perpendicular to each other, so the intersection of the diagonals forms a right angle and bisects the diagonals.

Figure 12: Kite square

If you can’t remember exactly what you mean by **symmetry** and the **axis of symmetry** imagine, we have a little repetition for you here:

Under **symmetry** is understood to mean a geometric property in which a figure forms a mirror image on both sides of an axis. As a result, symmetry is also referred to as «mirror image equality».

The axis or straight line that divides the figure into mirror images is called the mirror axis or **axis of symmetry** designated.

So that you can also calculate the perimeter (U) and the area (A) of a kite, it is best to learn these two formulas:

## The House of Squares

You have now learned about the different types of quadrilaterals and their properties and your head is probably spinning with all the definitions and formulas. To get an overview of the many squares, you can use the so-called** house of squares** on:

Figure 13: House of Squares

As you can see from the picture, the house of squares consists of different floors. The different squares are arranged according to their different properties and special features. At the lowest level, the ground floor, is the general quadrilateral. This has no special properties other than the four corners and four sides. The higher you go, the more specific properties you can discover in each type of square.

The categories that govern the classification of the various quadrilaterals are primarily the angular and lateral relationships and the symmetry properties. You may be wondering how the different squares are arranged within the floors and between the ground floor and the roof. We will try to explain this to you step by step:

FloorSquareTypeExplanationAtticSquare

**Peculiarities of the square:**

- four equal right angles
- all sides are the same length
- opposite sides are parallel to each other

3rd floor rhombus, rectangle**Characteristics of the rhombus:**

- four equal sides
- opposite sides are parallel to each other

**Peculiarities of the rectangle:**

- all four angles are equal
- opposite sides are equal in length and parallel to each other

2nd floorSquare kite, parallelogram, symmetrical trapezoid

**Properties of the kite square:**

- two pairs of sides that are equal in length
- two of the angles within the kite are equal

**Special features of the parallelogram:**

- the opposite sides are equal in size and parallel to each other
- opposite angles are equal

**Properties of the symmetrical trapezoid:**

- two equal sides
- two parallel sides
- two of the interior angles are equal

1st floor general trapezoid

**Special feature of the general trapezoid:**

- two parallel sides

First floorgeneral quadrilateralNo special properties of the general quadrilateral

## Square – The most important thing

- Quadrangle = 4 sides and 4 corners
- Differentiation between general and special quadrilaterals
- The classification is basically based on special properties, such as angles. symmetry, parallelism
- Great importance for all manual activities