# Square And Rectangle

Editor note: This blog post was originally published on 23 July 2018

## There is always plenty of buzzy stuff to discuss in maths especially when it comes to misconceptions, muddles and mix-ups. It’s important to tackle maths misconceptions head-on.

The three special parallelograms — rhombus, rectangle, and square — are so-called because they’re special cases of the parallelogram. (In addition, the square is a special case or type of both the rectangle and the rhombus.) The three-level hierarchy you see with in. A square has equal sides (marked 's') and every angle is a right angle (90°) Also opposite sides are parallel. A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

In fact, this is one of the recommendations in the ‘Improving Mathematics at Key Stages Two and Three‘ report from 2018. Simply pleading ignorance and continuing to teach them is no longer an option. Instead, we can positively use these misunderstandings and misconstructions during formative assessment, enabling us to help pupils “develop richer and more robust conceptions“.

## It’s time to go back to square one

The relationship between a square and a rectangle is 'type of'. A square is a type of rectangle, but a rectangle is not a type of square. I'm not aware of a single word that means 'type of'. In engineering and programming circles, this relationship is also described as 'is a'. Square and Rectangular A knitting pattern by Staci Perry, www.verypink.com Getting Started. Size: any size square, mine are about 9x9” Needles: size 7 US, straights or circulars Yarn: about 35 grams of worsted weight cotton (65 yards) Additional Materials: tapestry needle for weaving in ends.

The real problem with misconceptions is that when they’re unwittingly taught, children are being told faulty facts. Take the square, one of the first 2D shapes we’re taught, and something you’d imagine there wouldn’t be much confusion around. But how well do we really know it? Here’s the thing, a ‘square’ isn’t really a square. It’s actually a lazy label we have attached to it and, like super glue, it’s hard to shift.

When we say ‘square’ we use it as a noun when in reality ‘square’ is an adjective that describes a type of rectangle. What we should be telling children is that this shape is ‘a square rectangle’. It’s illuminating to see how prevalent the ‘square’ misconception is and that maths learners young and old are often taught to categorise rectangles and squares separately.

Pupils should be learning that a square is a more specific classification of a rectangle just as a rectangle is a more specific classification of a parallelogram, and a parallelogram is a specific classification of a quadrilateral.

## Why misconceptions become entrenched

Children have a firm and fixed view of shapes and the stereotypes they have are deeply entrenched and remarkably persistent. The reason why is simple. It’s because we’ve taught them.

But, as teachers, we can’t take all the blame.

Respected maths dictionaries, trusted resource books, gold standard websites and maths posters all play a part and are long overdue for an upgrade. If you have a shapes poster in your class, see how it conforms to stereotypical images and narrow definitions of shapes. A square rectangle is labelled a ‘square’, an oblong rectangle is just a ‘rectangle’, a parallelogram is always pictured as a pushed over rectangle, and so on. It’s time to challenge these traditional images and refine our own shape definitions so that children are getting correct information from the beginning. It’s time for a maths makeover!

In most maths classrooms, children learn the names for shapes without considering their essential properties. For instance, show colleagues or your class a ‘square rectangle’ and ask them to name it. Most will automatically say ‘square’. You might get some who respond with ‘quadrilateral’, ‘polygon’, ‘rectangle’ and ‘parallelogram’. A few might even venture a ‘rhombus’, ‘tetragon’ or ‘quadrangle’ but this will be rare. If you place the shape so it rests on one of its vertices then ‘diamond’ is sure to come up, as changing the orientation of a shape can dramatically alter our perception of it.

## Discussing misconceptions

Maths talk is vital to the process of judging the quality and depth of your pupils thinking. Why not try looking at some maths statements in small groups and getting your pupils to place each one into a column: either true, false, or it depends.

• A rectangle has four lines of symmetry
• A square is half the size of a rectangle
• An oblong is another name for a rectangle
• A rectangle has four congruent sides
• Every square is a rectangle
• Every rectangle is a square
• The diagonals of a rectangle cross at right angles
• A rectangle has rotational symmetry order of 4
• A parallelogram is a pushed over rectangle

Well-chosen true false statements are excellent training devices to develop mathematical thinking at a higher level. Their purpose is to challenge stereotypes and achieve mathematical insights based on conjecture and proof. They give pupils a challenge and promote thinking, disagreement and dollops of discussion when facilitated by expert teacher scaffolding.

Once children have articulated their ideas, you can listen and share them as a whole class and debate any similarities and differences. This ‘unpacking’ process is so important to our maths work and is recommended by Jeremy Hogden and Dylan Wiliam in their book ‘Mathematics Inside The Black Box’. It shows what children know, don’t know and partly know. If we ignore this way of working we could miss what is being said.

## Maths facts

What children talk about usually dictates which direction you move in, but providing definitions to support them and help shape their thinking and progress their ideas is also crucial. When teaching squares we need to set the record straight. By letting pupils know that there are two types of rectangle — square and non-square (oblong) — we can start to mathematise with more insight and accuracy.

The best maths teaching helps children to think about ‘rectangle‘ as the family name and ‘square‘ or ‘oblong‘ as the first name. Rectangle refers to any quadrilateral shape whose corners are all right-angled, opposite sides are equal and parallel and its diagonals bisect each other. A square rectangle is all those things with a couple of extra bits: all four sides are equal and its diagonals cross at right-angles.

So a square is definitely a rectangle but it is equilateral and equiangular too. All the other rectangles are non-square rectangles because they have one pair of sides longer than the other. These are oblong rectangles. A rectangle can be tall and thin, short and fat or all the sides can have the same length. So, a square is a special kind of rectangle.

Square‘ is an adjective to describe the type of rectangle so separating squares and rectangles and making them seem different is wobbly thinking. How is a rectangle different to a square? It isn’t. A square is a rectangle.

## Getting maths definitions right

The real issue is that children in primary classrooms do little more than learn the names of shapes rather than identifying properties. Their maths diet has been lacking.

In order to develop richer understanding in our pupils we need to do more. Why not get your class to design a wanted poster for a square rectangle, or write the definition of a square rectangle for a class maths dictionary. You could even ask the children to write to a dictionary company to get their definition. Children have recently got the definition of “bullying” changed in the Oxford English Dictionary so why can’t your pupils do the same for ‘square‘?

If our children are going to think independently and deeply about their maths we need to get the basics right first and challenge the way we have previously taught. Only then can we be sure that our pupils are learning correctly.

In Euclidean geometry, a quadrilateral is a four-sided 2D figure whose sum of internal angles is 360°. The word quadrilateral is derived from two Latin words ‘quadri’ and ‘latus’ meaning four and side respectively. Therefore, identifying the properties of quadrilaterals is important when trying to distinguish them from other polygons.

So, what are the properties of quadrilaterals? There are two properties of quadrilaterals:

• A quadrilateral should be closed shape with 4 sides
• All the internal angles of a quadrilateral sum up to 360°

This is what you’ll read in the article:

• Rectangle
• Square
• Parallelogram
• Rhombus
• Trapezium

Here is a video explaining the properties of quadrilaterals:

The diagram given below shows a quadrilateral ABCD and the sum of its internal angles. All the internal angles sum up to 360°.

Thus, ∠A + ∠B + ∠C + ∠D = 360°

There are 5 types of quadrilaterals on the basis of their shape. These 5 quadrilaterals are:

1. Rectangle
2. Square
3. Parallelogram
4. Rhombus
5. Trapezium

Let’s discuss each of these 5 quadrilaterals in detail:

## Rectangle

A rectangle is a quadrilateral with four right angles. Thus, all the angles in a rectangle are equal (360°/4 = 90°). Moreover, the opposite sides of a rectangle are parallel and equal, and diagonals bisect each other.

### Properties of rectangles

A rectangle has three properties:

• All the angles of a rectangle are 90°
• Opposite sides of a rectangle are equal and Parallel
• Diagonals of a rectangle bisect each other

### Rectangle formula – Area and perimeter of a rectangle

If the length of the rectangle is L and breadth is B then,

1. Area of a rectangle = Length × Breadth or L × B
2. Perimeter of rectangle = 2 × (L + B)

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## Square

Square is a quadrilateral with four equal sides and angles. It’s also a regular quadrilateral as both its sides and angles are equal. Just like a rectangle, a square has four angles of 90° each. It can also be seen as a rectangle whose two adjacent sides are equal.

### Properties of a square

For a quadrilateral to be a square, it has to have certain properties. Here are the three properties of squares:

• All the angles of a square are 90°
• All sides of a square are equal and parallel to each other
• Diagonals bisect each other perpendicularly

### Square formula – Area and perimeter of a square

If the side of a square is ‘a’ then,

1. Area of the square = a × a = a²
2. Perimeter of the square = 2 × (a + a) = 4a

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## Parallelogram

A parallelogram, as the name suggests, is a simple quadrilateral whose opposite sides are parallel. Thus, it has two pairs of parallel sides. Moreover, the opposite angles in a parallelogram are equal and its diagonals bisect each other.

### Properties of parallelogram

A quadrilateral satisfying the below-mentioned properties will be classified as a parallelogram. A parallelogram has four properties:

• Opposite angles are equal
• Opposite sides are equal and parallel
• Diagonals bisect each other
• Sum of any two adjacent angles is 180°

### Parallelogram formulas – Area and perimeter of a parallelogram

If the length of a parallelogram is ‘l’, breadth is ‘b’ and height is ‘h’ then:

1. Perimeter of parallelogram= 2 × (l + b)
2. Area of the parallelogram = l × h

## Rhombus

A rhombus is a quadrilateral whose all four sides are equal in length and opposite sides are parallel to each other. However, the angles are not equal to 90°. A rhombus with right angles would become a square. Another name for rhombus is ‘diamond’ as it looks similar to the diamond suit in playing cards.

### Properties of rhombus

A rhombus is a quadrilateral which has the following four properties:

• Opposite angles are equal
• All sides are equal and, opposite sides are parallel to each other
• Diagonals bisect each other perpendicularly
• Sum of any two adjacent angles is 180°

### Rhombus formulas – Area and perimeter of a rhombus

If the side of a rhombus is a then, perimeter of a rhombus = 4a

If the length of two diagonals of the rhombus is d1 and d2 then the area of a rhombus = ½ × d1 × d2

## Trapezium

A trapezium (called Trapezoid in the US) is a quadrilateral which has only one pair of parallel sides. The parallel sides are referred to as ‘bases’ and the other two sides are called ‘legs’ or lateral sides.

### Properties of Trapezium

A trapezium is a quadrilateral in which the following one property:

• Only one pair of opposite sides are parallel to each other

### Trapezium formulas – Area and perimeter of a trapezium

If the height of a trapezium is ‘h’ (as shown in the above diagram) then:

1. Perimeter of the trapezium= Sum of lengths of all the sides = AB + BC + CD + DA
2. Area of the trapezium = ½ × (Sum of lengths of parallel sides) × h = ½ × (AB + CD) × h

The below table summarizes all the properties of the quadrilaterals that we have learned so far:

The below image also summarizes the properties of quadrilaterals:

The below table summarizes the formulas on area and perimeter of different types of quadrilaterals:

Let’s practice the application of properties of quadrilaterals on the following sample questions:

### Question 1

Adam wants to build a fence around his rectangular garden of length 10 meters and width 15 meters. How many meters of the fence he should buy to fence the entire garden?

1. 20 meters
2. 25 meters
3. 30 meters
4. 40 meters
5. 50 meters
##### Solution

Step 1: Given

• Adam has a rectangular garden.
• It has a length of 10 meters and a width of 15 meters.
• He wants to build a fence around it.

Step 2: To find

• The length required to build the fence around the entire garden.

Step 3: Approach and Working out

The fence can only be built around the outside sides of the garden.

• So, the total length of the fence required= Sum of lengths of all the sides of the garden.
• Since the garden is rectangular, the sum of the length of all the sides is nothing but the perimeter of the garden.
• Perimeter = 2 × (10 + 15) = 50 metres

Hence, the required length of the fence is 50 meters.

Therefore, option E is the correct answer.

### Question: 2

Steve wants to paint one rectangular-shaped wall of his room. The cost to paint the wall is \$1.5 per square meter. If the wall is 25 meters long and 18 meters wide, then what is the total cost to paint the wall?

## Square And Rectangle

1. \$ 300
2. \$ 350
3. \$ 450
4. \$ 600
5. \$ 675
##### Solution

Step 1: Given

• Steve wants to paint one wall of his room.
• The wall is 25 meters long and 18 meters wide.
• Cost to paint the wall is \$1.5 per square meter.

Step 2: To find

• The total cost to paint the wall.

Step 3: Approach and Working out

• A wall is painted across its entire area.
• So, if we find the total area of the wall in square meters and multiply it by the cost to paint 1 square meter of the wall then we can the total cost.
• Area of the wall = length × Breadth = 25 metres × 18 metres = 450 square metre
• Total cost to paint the wall = 450 × \$1.5 = \$675

Hence, the correct answer is option E.

We hope by now you would have learned the different types of quadrilaterals, their properties, and formulas and how to apply these concepts to solve questions on quadrilaterals. The application of quadrilaterals is important to solve geometry questions on the GMAT. If you are planning to take the GMAT, we can help you with high-quality study material which you can access for free by registering here.

## Square And Rectangle Shape

Here are a few more articles on Math:

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## FAQs

What are the different types of quadrilaterals?

There are 5 types of quadrilaterals – Rectangle, Square, Parallelogram, Trapezium or Trapezoid, and Rhombus.

Where can I find a few practice questions on quadrilaterals?