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• Xuming Liang
• Christianna Rodriguez
• Sandeep Bhardwaj
• Agnishom Chattopadhyay
• Zandra Vinegar
• Andrew Hayes

Symmetry describes when several parts of an object are identical, such that it's possible to flip, spin, and/or move the object without ultimately changing what it looks like. Symmetry is extremely powerful and beautiful problem-solving tool and it appears all over the place: in art, architecture, nature, and all fields of mathematics! The three basic kinds of 2-dimensional symmetry are reflection , rotation , and translation . After having fun reading through this page, you will be able to recognize all forms of 2-dimensional symmetry.

In fact, soon you won't be able to stop seeing symmetry! It's everywhere, even in the letters on this page for example. The above are a few letters in the English alphabet that have different styles of symmetry (a rotation and two different reflections). We will explore all of the beautiful kinds of symmetry in detail below.

Reflections

Translations, combining symmetries.

Rotational symmetry can be seen in some of the most beautiful aspects of nature, art, mathematics, and practical, man-made objects. Here, we will define rotational symmetry, describe the "orders" that rotational symmetry can have, and discuss several interesting examples of rotational symmetry.

Rotational symmetry: An object has rotational symmetry if it looks unchanged after being turned (rotated) by some specific amount less than a full 360-degree spin. The center-point of the spin is called the point of symmetry of the rotation.

Let's look at some of the letters in the alphabet which have rotational symmetry.

Turn any book that you're reading upside-down (a 180-degree rotation)--many letters like S and Z will be 'unaffected.'

Our next step will be to examine the order of rotational symmetry of an object. Let's start with a formal definition of the order of rotational symmetry.

Order of Rotational Symmetry When an object has rotational symmetry, the order of that symmetry is the number of different positions in which, after some precise amount of rotation, the object looks exactly the same as it did originally.

For example, look at the following diagram:

So the rotational symmetry of an object can have order 2, 3, 4, 5, 6, ..., or any other whole number \(n.\) Except... did we forget to mention the example of rotational symmetry of order 1? No, we did not, and this is intentional. The rotational symmetry of order 1 does not exist since if a shape only matches itself once as we go around (i.e. it matches itself after one full rotation), then there is really no symmetry at all.

What is the order of rotational symmetry of the world's tallest Ferris wheel, also known as the London Eye? The world's tallest Ferris wheel London Eye has a rotational symmetry of order 32. \(_\square\)

What is the order of rotational symmetry possessed by the parallelogram shown below? In a parallelogram, the opposite sides are parallel and its diagonals bisect each other. We will rotate the parallelogram by \(90^\circ\) in each successive turn till one complete revolution and then see how many times it has attained the same look as the original one. The parallelogram attained the same look two times in one complete rotation, i.e. one after \(180^\circ\) rotation and another after \(360^\circ\) rotation. So the order of rotational symmetry of the parallelogram is also 2. \(_\square\)

Can you find the order of rotational symmetry of the beautiful flower shown below? It looks the same five times when rotated through a complete cycle. Hence, the order of rotational symmetry of the flower is 5. \(_\square\)

Does the Brilliant logo possess rotational symmetry? If yes, then what is its order? Let's take the curved triangle as our focal point. Since the triangle is curved unevenly, things will look different from each side. Therefore, the Brilliant logo does not possess rotational symmetry. \(_\square\)

As we have gone through several examples of rotational symmetry, here are some problems based on the same concept:

Find the order of rotational symmetry of the octagon given above.

Does the US Bronze Star Medal have rotational symmetry? If so, then what is the order of symmetry?

For the purposes of this problem, ignore the striped red ribbon as part of the medal.

We have gone through several interesting scenarios involving rotational symmetry. To continue our journey, now we will move on to discuss reflection symmetry and the many places where it appears. We will start by defining reflection symmetry.

Reflection symmetry: An object has reflection symmetry if there exists a line that the object can be flipped (reflected) over such that it looks exactly the same before and after the flip. The line that the object flips across is called the axis of symmetry (plural: axes of symmetry) .

An object may have 0, 1 or more than one axes of symmetry. An axis of symmetry is also sometimes called a line of symmetry.

For example, the letters A and M, each have a vertical axis of symmetry, the letters E and K both have horizontal axes of symmetry, and the letters H and X have both horizontal and vertical axes of symmetry.

How do we know whether any line is an axis of symmetry for an object or not? It is very simple: an axis of symmetry essentially behaves like a mirror. In place of the axis, if we place a mirror, then we should get the same shape. Let's look at an example.

Discuss the symmetry of an Isosceles triangle. Isosceles Triangles We see that if we draw a vertical line in the center, then the triangle will be divided into two parts, which are reflections of each other. Therefore an isosceles triangle has reflection symmetry. \(_\square\) Note: How do we recognize an axis of symmetry in a given object? Well, it's a skill, and it only gets better with practice!

Many geometrical figures have one or more axes of symmetry: squares, rectangles, parabolas, hyperbolas, circles, ellipses, etc. (The list is endless!) But it's easy to be tempted to see extra axes of symmetry. For example, here is a question that most people get wrong if they don't consider it carefully!

Is the diagonal of a rectangle an axis of symmetry? A common misconception is that all lines that divide a figure into two congruent parts are lines of symmetry. This is not always the case! For example, the diagonal of a (non-square) rectangle is not a line of symmetry. \(_\square\) Note: If you actually reflect the lower-left half of a rectangle over the diagonal, the figure you form is called a kite. \(\quad \quad\)

How many lines of symmetry does this regular 5-pointed star have? The 5-pointed regular star has 5 lines of symmetry. We can see them clearly in the following figure. There is a line of symmetry passing through each of the 5 points of the star. \(_\square\)

The amazing fact that reflection symmetry is abundant in nature as well compels us to love symmetry and we can have fun with it. Here is an interesting fact: More than 99% of the world's animals have reflection symmetry, including the familiar homo sapien species!

Show that the human body possesses reflection symmetry. From the outside, at first sight the human body at a frontward facing angle appears to be symmetrical. We can draw a vertical axis of symmetry in the center. \(_\square\)

The image of an orchid flower you see above also has reflection symmetry.

Many man-made structures also exhibit reflection symmetry. Symmetry looks beautiful and appeals the eye. It's no wonder so many famous structures are symmetrical. Let's take a tour around the world here through the following examples:

One of the seven wonders of the world: Taj Mahal The beautiful Adelaide Studios Petronas Twin Towers

Here are a few problems for you to try yourself, which will check your understanding of the axes of symmetry.

The above picture shows a beautiful example of fractal flame which is a type of fractal art. The image is aptly titled "Sunflower Glory."

How many axes of symmetry are present in the image "Sunflower Glory"?

Image Credit

How many of the following 8 shapes must have (at least) one axis of symmetry?

\[\begin{array}&&\text{equilateral triangle}, &\text{right triangle}, &\text{square}, &\text{cyclic quadrilateral}, \\ &\mbox{parallelogram}, &\text{rhombus}, &\text{rectangle}, &\text{regular pentagon} \end{array} \]

We can now easily figure out the symmetries present in an object or figure. Let's work out an example and a problem to grab our understanding about the number of lines of symmetry.

How many lines of symmetry does the Mercedes-Benz logo have? Try counting the number of lines of symmetry without looking at the circle . If the circle wasn't there, and we just had three lines extending from one point, we can easily figure out that there are \(3\) lines of symmetry. \(_\square\)

Consider a regular \(n\)-gon, where \(n\) is an even integer. How many lines of symmetry will this shape have? There will be several different lines to count: There are already \(\frac{n}{2}\) lines of symmetry, each one passing through one side and leaving from the opposite. So as not to double count, we divide the number of sides by \(2\). We will also have another \(\frac{n}{2}\) lines, each one going through one of the \(n\)-gon's corners and leaving from the opposite. Thus, an \(n\)-gon with an even \(n\) has \(n\) lines of reflective symmetry. \(_\square\)

Consider a circle divided up into 11 equal sections. How many lines of symmetry will this circle have?

The amazing fact is that the snowflakes too possess symmetries. Let's get more familiar with the snowflakes through examples and problems.

Discuss the symmetry of the snowflakes. All snowflakes have rotational symmetry of order 6. Snowflakes are symmetrical because they reflect the internal order of the water molecules as they arrange themselves in the solid state (the process of crystallization). Water molecules in the solid state, such as in ice and snow, form weak bonds (called hydrogen bonds) to one another. These ordered arrangements result in the basic symmetrical, hexagonal shape of the snowflake. \(_\square\)

When he was 20, Wilson Alwyn “Snowflake” Bentley photographed his first snowflake by perfecting the process of catching the flakes on black velvet before they melted or sublimated. He pioneered the art of snowflake photography, and captured over 5000 beautiful images for us.

In the above image, how many lines of symmetry does the snowflake have?

Wow! Through the sections above, we found that we are surrounded by symmetrical objects, even many living ones! We still have one further kind of symmetry to discuss: translational symmetry . Let's start with the definition of translational symmetry and then we will move on to visualizing the symmetry and playing around with it.

Translational symmetry: Translational symmetry is when a pattern looks the same before and after it has been moved ("translated") a particular distance in a specified direction. Patterns with translational symmetries look like they were created by someone with a stamp, moving in a straight line across a surface and stamping at regular intervals as they move. An example is as follows:

\(\large \color{green}{\text{This wiki page is really awesome!}}\) \(\large \quad \quad \quad \color{green}{\text{This wiki page is really awesome!}}\) \(\large \quad \quad \quad \quad \quad \quad \color{green}{\text{This wiki page is really awesome!}}\)

An object or pattern has translational symmetry if it can be slid in some direction and be ultimately unchanged at the end of the motion. For example, an infinitely ongoing series of ellipses (.........) has translational symmetry.

Let's work out some interesting examples with translational symmetry and have fun exploring them.

The graph of \(\text{CO}_2\) in ppm (parts per million) has translational symmetry as you can see in the above graph. The reason behind it is as follows: Carbon dioxide \((\text{CO}_2)\) is an important heat-trapping (greenhouse) gas, which is released through human activities such as deforestation and burning fossil fuels, as well as natural processes such as respiration and volcanic eruptions. The seasonal cycle is due to the vast land mass of the Northern Hemisphere, which contains the majority of land-based vegetation. The result is a decrease in atmospheric carbon dioxide during northern spring and summer when plants are absorbing \(\text{CO}_2\) as part of photosynthesis. The pattern reverses, with an increase in atmospheric carbon dioxide during northern fall and winter. The yearly spikes during the cold months occur as annual vegetation dies and leaves fall and decompose, which releases their carbon back into the air.

There are several man-made objects which possess translational symmetry. Here we are listing a few of them, but you are free to explore the objects possessing translational symmetry and have fun with them.

Street Lamps Train Tracks Designing Wallpaper

Here's one special case of transnational symmetry: a combination of sorts between a translation and a reflection called a glide reflection .

Glide reflection: A glide reflection is a combination of two transformations: a reflection over a line followed by a translation in the same direction as the line.

Only an infinite strip can have translation symmetry or glide reflection symmetry. For translation symmetry, you can slide the whole strip some distance, and the pattern will land back on itself. For glide reflection symmetry, you can reflect the pattern over some line, then slide in the direction of that line, and it looks unchanged. And the patterns must go on forever in both directions.

For an example, see the following:

Main article: Frieze Patterns

We have gone through all the symmetry types one by one, and now let's apply different types of symmetry in the same problem!

Let's start by working through some interesting examples which use both reflectional and rotational symmetries, but no translational symmetry.

M. C. Escher We will end this page by mentioning one amazing artist who spent much of his life exploring symmetries and creating stunning designs that combine an advanced understanding of the mathematics involved with awe-inspiring creativity! M. C. Escher was a Dutch mathematician who dedicated most of his life to studying the mathematics of tessellations . He created drawings which demonstrate the beauty of mathematics through geometry and patterns.

Aperiodic Penrose Tiling A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles, named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s. The aperiodicity of the Penrose tiling implies no translational symmetry, i.e. a shifted copy of one will never match the original. A Penrose tiling may be made with both reflection symmetry and rotational symmetry of order 5, as in the example below.

Discuss the different symmetries in a circle. Reflection symmetry: There is an infinite number of lines of reflection symmetry in a circle. Rotational symmetry: There are also an infinite number of orders of rotational symmetry since you can turn a circle any amount and it will remain unchanged. \(_\square\)

Just one more problem and we will be accomplishing all our objectives of going through this page. Here we go:

A person paints 3 small triangles in the top half of the figure \(\color{blue}{\text{blue}}\). He then paints 5 triangles in each half \(\color{green}{\text{green}}\), leaving 8 in each half remain \(\color{darkorange}{\text{orange}}\). When the upper half is folded down over the center line, 2 pairs of \(\color{blue}{\text{blue}}\) triangles coincide as do 3 pairs of \(\color{green}{\text{green}}\) triangles. There are also 2 \(\color{blue}{\text{blue}}\)-\(\color{darkorange}{\text{orange}}\) pairs. How many \(\color{darkorange}{\text{orange}}\) pairs coincide? There are total 16 small triangles in each half and each half has 3 \(\color{blue}{\text{blue}}\) triangles, 5 \(\color{green}{\text{green}}\) triangles, and 8 \(\color{darkorange}{\text{orange}}\) triangles. There are also 2 pairs of \(\color{blue}{\text{blue}}\) triangles, so 2 \(\color{blue}{\text{blue}}\) triangles on each side are used, leaving 1 \(\color{blue}{\text{blue}}\) triangle, 5 \(\color{green}{\text{green}}\) triangles, and 8 \(\color{darkorange}{\text{orange}}\) triangles remaining on each half. Also, there are 3 pairs of \(\color{green}{\text{green}}\) triangles, using 3 \(\color{green}{\text{green}}\) triangles on each side, so there is 1 \(\color{blue}{\text{blue}}\) triangle, 2 \(\color{green}{\text{green}}\) triangles, and 8 \(\color{darkorange}{\text{orange}}\) triangles remaining on each half. Also, we have 2 \(\color{blue}{\text{blue}}\)-\(\color{darkorange}{\text{orange}}\) pairs. This obviously can't use 2 \(\color{blue}{\text{blue}}\) triangles on one side, since there is only 1 on each side, so we must use 1 \(\color{blue}{\text{blue}}\) triangle and 1 \(\color{darkorange}{\text{orange}}\) triangle per side, leaving 2 \(\color{green}{\text{green}}\) triangles and 7 \(\color{darkorange}{\text{orange}}\) triangles apiece. The remaining \(\color{green}{\text{green}}\) triangles cannot be matched with other \(\color{green}{\text{green}}\) triangles since that would mean there were more than 3 \(\color{green}{\text{green}}\) pairs, so the remaining \(\color{green}{\text{green}}\) triangles must be paired with \(\color{darkorange}{\text{orange}}\) triangles, yielding 4 \(\color{green}{\text{green}}\)-\(\color{darkorange}{\text{orange}}\) pairs, one for each of the remaining \(\color{green}{\text{green}}\) triangles. This uses 2 \(\color{green}{\text{green}}\) triangles and 2 \(\color{darkorange}{\text{orange}}\) triangles on each side, leaving 5 \(\color{darkorange}{\text{orange}}\) triangles apiece, which must be paired with each other, so there are 5 \(\color{darkorange}{\text{orange}}\)-\(\color{darkorange}{\text{orange}}\) pairs. \(_\square\)

Let's accomplish the objectives of this page by solving the following problem:

The figure at right possesses which of the following symmetry/symmetries?

I. Rotational symmetry II. Translational symmetry III. Reflection symmetry

Problem Loading...

Note Loading...

Set Loading...

• 1 Hidden symmetry
• 2 Symmetry with respect angle bisectors
• 3 Symmetry with respect angle bisectors 1
• 4 Construction of triangle
• 5 Symmetry with respect angle bisectors 2
• 6 Symmetry of radical axes
• 7 Composition of symmetries
• 8 Composition of symmetries 1
• 9 Composition of symmetries 2
• 10 Symmetry and secant
• 11 Symmetry and incircle
• 12 Symmetry and incircle A
• 13 Symmetry for 60 degrees angle
• 14 See also

Hidden symmetry

Symmetry with respect angle bisectors

Symmetry with respect angle bisectors 1

[email protected], vvsss

Construction of triangle

Construction

Symmetry with respect angle bisectors 2

Symmetry of radical axes

Composition of symmetries

It is known that the composition of two axial symmetries with non-parallel axes is a rotation centered at point of intersection of the axes at twice the angle from the axis of the first symmetry to the axis of the second symmetry.

Composition of symmetries 1

Composition of symmetries 2

Symmetry and secant

Symmetry and incircle

Symmetry and incircle A

Symmetry for 60 degrees angle

a) One can find successively angles (see diagram).

• Symmedian and Antiparallel

Something appears to not have loaded correctly.

Click to refresh .

Symmetry Questions

Symmetry Questions are a set of questions based on the topic of symmetry. These questions help the student in understanding the topic better. It also improves the spatial sense of the student and improves the thinking ability which is due to the imagination needed to solve the problems. Before answering the questions, for detailed information, visit symmetry .

For Synopsis: Symmetry is a wide topic that is specific to shapes and their appearances. Symmetry is a geometrical topic, where a shape is observed to have an imaginary line called a line of axis. This line of axis divides the shape in such a way that both the left side and right side of the line of axis are a mirror images to each other. Let us consider a simple example of the letter M. Note that M is also a shape obtained by straight lines. Now M can be split in a way such that both halves look like the mirror image of one another.

Topics to refer before answering Symmetry Questions are

• Lines of Symmetry :

The above example has only one line of symmetry. There are shapes which can have multiple lines of symmetry. Ex. Circle can have an infinite number of symmetrical lines.

• Rotation Symmetry.

When a shape is moved or rotated, if the shape appears the same, then the shape has a rotational symmetry

• Reflection Symmetry

Symmetry which has a mirror reflection. Same as letter M.

• Translational Symmetry

When an image is moved/displaced from its original position, but if the orientation remains the same.

• Glide Symmetry

When an image undergoes both Translational and reflection symmetry then we call it as glide symmetry

Let us start solving the Symmetry Questions.

Symmetry Questions With Solutions

Here are a few Symmetry questions with the answers.

Question 1:

How many English alphabets are symmetrical?

14 letters have lines of symmetry either horizontally or vertically.

They are A, B, C, D, E, H, I, K, M, O, T, V, W, X.

Question 2:

How many lines of symmetry does these geometrical shapes have?

• Square : has 4 lines of symmetry
• Pentagon: has 5 lines of symmetry
• Rhombus: has 2 lines of symmetry
• Trapezium: has 0 lines of symmetry

Question 3:

Which of the following shapes have rotational symmetry?

• A square has a rotational symmetry of order 4
• Circle has a rotational symmetry at every angle of the circle
• Hexagon has rotational symmetry of order 6
• Crescent has rotational symmetry of order 2

Question 4:

Draw the lines of Symmetry for the following shapes.

The Line of Symmetry for the given shapes are:

Question 5:

Show translation symmetry to the following images

• Square has been translated from the 1st Quadrant to the 3rd Quadrant.
• Hexagon Translated from the 2nd Quadrant to 3rd Quadrant
• Triangle translated from 2nd Quadrant to 4th Quadrant.

Question 6:

How do reflection symmetry look for the following shapes?

The reflection for the Star, Moon and Sun appear the same as the original images

Question 7:

Draw a shape that does not have symmetry.

Question 8:

Does the image have a symmetry along x- axis?

The given image does not have symmetry along x- axis. However it is symmetrical diagonally.

Question 9:

How to make the given image symmetrical along the y-axis?

Step 1: Remove the cell in the 1st row, 3rd column

Step 2: Add the 5th cell in the 1st column in the place of the 2nd column fourth row.

Question 10:

Colour the appropriate cells to make it symmetrical for the given image.

There are multiple ways to make the given image symmetrical. 3 methods are given below.

Colour all cells with the same colour. (either grey or white) to make it symmetrical

Video Lesson on Symmetry

Related Articles:

• Line of Symmetry
• Figures with Symmetry
• Reflection and Symmetry
• Rotational Symmetry

Practice Questions on Symmetry

• Draw a geometrical figure with no Symmetry.
• What is the rotational symmetry of a nonagon?

• What is the reflection symmetry for the letter M.

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• DOI: 10.1080/00207390050203315
• Corpus ID: 119911851

Applications of symmetry to problem solving

• R. Leikin , A. Berman , O. Zaslavsky
• Published 1 November 2000
• Mathematics, Education
• International Journal of Mathematical Education in Science and Technology

27 Citations

Improving problem solving by exploiting the concept of symmetry, an exploratory study on students reasoning about symmetry, symmetry in mathematics and art: an exploration of an art venue for mathematics learning, problem-solving preferences of mathematics teachers: focusing on symmetry, teaching of symmetry at mathematics lessons in the first forms of azerbaijan primary schools, mathematical walks in search of symmetries: from visualization to conceptualization, prospective teachers' development of meta-cognitive functions in solving mathematical-based programming problems with scratch, symmetry and rotation skills of prospective elementary mathematics teachers.

• Highly Influenced

The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences

Axial symmetry in primary school through a milieu based on visual programming, 11 references, symmetry in science : an introduction to the general theory, teaching and learning mathematical problem solving : multiple research perspectives, problem-solving strategies, geometric transformations i: symmetry, symmetry 2 : unifying human understanding, mathematical problem solving, problem solving in the mathematics curriculum. a report, recommendations, and an annotated bibliography. maa notes, number 1., related papers.

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Applications of symmetry to problem solving

• Teaching and Learning

Research output : Contribution to journal › Article › peer-review

Symmetry is an important mathematical concept which plays an extremely important role as a problem-solving technique. Nevertheless, symmetry is rarely used in secondary school in solving mathematical problems. Several investigations demonstrate that secondary school mathematics teachers are not aware enough of the importance of this elegant problem-solving tool. In this paper we present examples of problems from several branches of mathematics that can be solved using different types of symmetry. Teachers' attitudes and beliefs regarding the use of symmetry in the solutions of these problems are discussed.

ASJC Scopus subject areas

• Mathematics (miscellaneous)
• Applied Mathematics

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• 10.1080/00207390050203315

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• Attitude and Beliefs Psychology 100%
• School Mathematics Psychology 100%
• Mathematical Problem Psychology 100%
• Problem-solving Tools Keyphrases 100%
• Secondary School Mathematics 100%
• Attitude Medicine and Dentistry 100%

T1 - Applications of symmetry to problem solving

AU - Leikin, Roza

AU - Berman, Abraham

AU - Zaslavsky, Orit

N2 - Symmetry is an important mathematical concept which plays an extremely important role as a problem-solving technique. Nevertheless, symmetry is rarely used in secondary school in solving mathematical problems. Several investigations demonstrate that secondary school mathematics teachers are not aware enough of the importance of this elegant problem-solving tool. In this paper we present examples of problems from several branches of mathematics that can be solved using different types of symmetry. Teachers' attitudes and beliefs regarding the use of symmetry in the solutions of these problems are discussed.

AB - Symmetry is an important mathematical concept which plays an extremely important role as a problem-solving technique. Nevertheless, symmetry is rarely used in secondary school in solving mathematical problems. Several investigations demonstrate that secondary school mathematics teachers are not aware enough of the importance of this elegant problem-solving tool. In this paper we present examples of problems from several branches of mathematics that can be solved using different types of symmetry. Teachers' attitudes and beliefs regarding the use of symmetry in the solutions of these problems are discussed.

UR - http://www.scopus.com/inward/record.url?scp=84857880954&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857880954&partnerID=8YFLogxK

U2 - 10.1080/00207390050203315

DO - 10.1080/00207390050203315

M3 - Article

AN - SCOPUS:84857880954

SN - 0020-739X

JO - International Journal of Mathematical Education in Science and Technology

JF - International Journal of Mathematical Education in Science and Technology

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Applications of symmetry to problem solving

2000, International Journal of Mathematical Education in Science and Technology

Related Papers

Mihriban HACISALİHOĞLU KARADENİZ

This study is an explanatory study, which follows a qualitative methodology and aims to reveal the explanations of prospective mathematics teachers for the potential misconceptions of secondary school students in relation to the concept of symmetry and the correction of such misconceptions. The study group consisted of a total of 26 prospective middle school mathematics teachers, who were senior students. The data of the study were obtained from the open-ended test prepared by the researchers. Following the data analysis, it was found out that the majority of the participants failed to identify the mistakes of students and suggested the ways to " letter the corners of the shape and measure the distance based on the axis of symmetry " in order to correct these mistakes. Moreover, the study notably observed that the participants adopted practical solutions such as the use of a mirror, paper folding, and the use of unit squares more in teaching the concept of symmetry. It may be stated that the prospective teachers, in this way, overlooked the development of conceptual knowledge in students.

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This paper describes Singapore and Australian Grade 6 students' (n=1187) performance on a symmetry task in a recently developed Mathematics Processing Instrument (MPI). The MPI comprised tasks sourced from Australia and Singapore's national assessments, NAPLAN and PSLE. Only half of the cohort solved the item successfully. It is possible that persistence of prototypical images of a vertical line of symmetry and reinforcements in the classroom could have contributed to this low success rate.

The concept of symmetry is vital for the study of science and developing spatial and geometric relationships. Yet few studies have been conducted to examine how students learn this concept. This study investigates 757 Year 4 to 10 students ability to reason about symmetrical relationships. By analysing the semiotic processes through the lens of the theory of semiotic mediation and Sfards mathematical discourse framework, the results show the synergic relations between visualisation of artefacts, and the use of signs and keywords in the construction of knowledge.

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Catholic University in Ruomberok, Slovakia and Pamukkale University in Denizli, Turkey prepare future primary teachers. Most of the students are women without an overly positive approach to mathematics and geometry in particular, yet in the future they will be teaching children in primary schools. According to the curricula in both countries symmetry plays an important role in the teaching of mathematics at kindergarten and primary levels. This paper shows a comparative-descriptive study of pre-service elementary teachers' ability for symmetry. We describe some results of this research along with recommendations for university teachers lecturing on kindergarten and primary education.

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Although the knowledge required by mathematics teacher educators is a relatively recent area of research, there has been significant progress in the field over the last few years. The classic distinction of a teacher’s knowledge into content knowledge and pedagogical content knowledge prompts us to reflect in this regard on what should constitute the content of primary teacher education programmes, and how the educator might mediate this content to make it accessible to their prospective teachers. This paper aims to contribute to this progress through a study into the work of Lucas, a teacher educator, during the course of a training session with prospective primary teachers. Critical observation of the video recording brought to the fore salient teaching situations on the topic of symmetry, which led us to explore the pedagogical content knowledge deployed in the session through a guided interview with the educator. Analysis of extracts from this interview enabled us to identify th...

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Symmetry – Definition, Examples, Practice Problems, FAQs

Line of symmetry, symmetry in real life , solved examples on symmetry, practice problems on symmetry, frequently asked questions on symmetry.

In mathematics, especially in geometry and its applications, an object is said to have symmetry if it can be divided into two identical halves. For example, look at the given picture of a flower:

If we were to draw an imaginary line in the middle of it, we could divide it into two equal parts like this:

Note that the two parts are identical and mirror images of each other.

An object that is not symmetric is said to be asymmetric. That means that an asymmetric object cannot be divided into identical halves. 

An example of asymmetry would be the given image.

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Let’s look at this flower again, the dotted line along which we divided it into two identical halves is called the line of symmetry or the axis of symmetry. It can also be defined as the line along which we can fold the object and its left and right half would completely overlap with each other. 

In mathematics, there are three types of lines of symmetry . 

• The Vertical Line of Symmetry

The Horizontal Line of Symmetry

The Diagonal Line of Symmetry

Let’s take a look at each of these lines.

Vertical Line of Symmetry

A vertical line that divides an object into two identical halves is called a vertical line of symmetry. That means that the vertical line goes from top to bottom (or vice versa) in an object and divides it into its mirror halves. For example, the star below shows a vertical line of symmetry.

When a horizontal line divides an object into two identical halves, it is called a horizontal line of symmetry. That means the horizontal line of symmetry goes from left to right (or vice versa) in an object. For example, the image below shows a horizontal line of symmetry.

When a diagonal / skew line divides an object into two identical halves, it is called a diagonal line of symmetry. For example, the square below shows a diagonal line of symmetry.

In fact, a square possesses all three lines of symmetry.

So we can say that an object can have multiple lines of symmetry. Some other examples of shapes that have multiple lines of symmetry are circles and rectangles . 

An example of a shape that does not have any line of symmetry is a scalene triangle . Since all sides of a scalene triangle are different, it cannot be divided into two identical mirror halves.

Symmetry was taught to humans by nature itself. A lot of flowers and most of the animals are symmetric in nature. Inspired by this, humans learned to build their architecture with symmetric aspects that made buildings balanced and proportionate in their foundation, like the pyramids of Egypt! We can observe symmetry around us in many forms:

• Trees reflected in crystal clear water and towering mountains reflected in a lake.
• The feathers of a peacock and the wings of butterflies and dragonflies have identical left and right sides.
• Hives of honeybees are made of hexagonal shape, which is symmetric in nature.
• Snowflakes in winter have all three lines of symmetry.

Example 1: How many lines of symmetry does the given figure have?

Answer: Only one line of symmetry.

Example 2: Can you find any examples of letters from the alphabet that have a horizontal line of symmetry?

Answer: A lot of letters, like O, D, H, have horizontal lines of symmetry.

Example 3: Is the given shape symmetric or asymmetric?

Answer: The given shape is symmetric with a vertical line of symmetry.

Attend this Quiz & Test your knowledge.

How many lines of symmetry does a rectangle have?

Which of the following letters of the alphabet have a horizontal and vertical line of symmetry?

Which line of symmetry does the following figure have.

Which of the following numbers has two lines of symmetry?

What is symmetry in math?

Symmetry in math is the property of an object to be able to be divided into two identical mirror halves.

What is the axis of symmetry?

The axis of symmetry is defined as the imaginary line along which an object can be folded or be divided into two identical mirror halves.

How many lines of symmetry are there?

There are three lines of symmetry, namely the horizontal line of symmetry, the vertical line of symmetry, and the diagonal line of symmetry.

How many lines of symmetry does a circle have?

A circle can be divided into two identical halves by folding it along its diameter. Since a circle has infinite diameters, it has infinite lines of symmetry.

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Symmetry Worksheets Line Symmetry Easier

Welcome to the Math Salamanders Line Symmetry Worksheets page. Here you will find a range of free printable symmetry worksheets, which will help your child to practice their reflecting and flipping skills.

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Symmetry Help

The Math Salamanders have a large bank of free printable symmetry worksheets. Each symmetry sheet comes complete with answers for support.

Handy Hints

Each point or block that has been reflected must remain the same distance from the mirror line as the original point. So if point A is 3 squares away from the mirror line, then the reflection of point A must also be 3 squares away.

When reflecting a shape, look at the corners of the shape and reflect each corner first as a dot in the mirror line. The dots can then be joined up (in the correct order!)

For lines of symmetry at angles of 45°, it is often better to rotate your paper so that the line of symmetry is vertical or horizontal, and the rest of the paper is at an angle.

The basis and understanding of symmetry starts at about Grade 2, and then develops further in Grades 3,4 and 5.

Line Symmetry Worksheets

On this webpage you will find our range of line symmetry sheets for kids.

The sheets have been carefully graded with the easier sheets coming first. The first 3 worksheets involve only horizontal and vertical lines only. The next 3 worksheets involve reflecting diagonal lines as well.

There are also some templates at the end of this section for you to create your own shapes for your child to reflect, or, even better, for your child to create their own symmetric patterns!

Using these sheets will help your child to:

• learn to reflect a shape in a vertical or horizontal mirror line;
• learn to reflect a shape in both a vertical and horizontal mirror line;

Reflecting in 1 mirror line

Horizontal and vertical lines only.

• Line Symmetry 1
• PDF version
• Line Symmetry 2
• Line Symmetry 3
• Line Symmetry 4
• Line Symmetry 5
• Line Symmetry 6

Reflecting in Diagonal mirror lines

• Line Symmetry 9
• Line Symmetry 10

Reflecting in vertical, horizontal and diagonal lines

• Line Symmetry 11

Reflecting in 2 mirror lines

• Line Symmetry 7
• Sheet 7 Answers
• Line Symmetry 8
• Sheet 8 Answers

Reflecting in vertical and horizontal or 2 diagonal lines

• Line Symmetry 12

Line Symmetry templates

• Line Symmetry Template 1
• Line Symmetry Template 2
• Line Symmetry Template 3
• Line Symmetry Template 4
• Line Symmetry Template 5
• Line Symmetry Template 6

Looking for something harder?

Here you will find a range of line symmetry activity sheets with one or two mirror lines.

The sheets in this section are similar to those on this page, but are more complicated and at a harder level.

• Harder Symmetry Activities

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Block Symmetry Worksheets

Here you will find a range of symmetry worksheets reflecting blocks instead of lines.

These sheets are at an easier level than the ones on this page.

• Symmetry Worksheets - Block Symmetry
• Explore 2d Shapes Worksheets

Looking for some geometry worksheets to get children thinking and reasoning about 2d shapes?

The shapes on this page are all about children really understanding what 2d shapes are all about, and using their reasoning skills to justify their thinking.

• know the properties of a range of 2d shapes;
• recognise that some shapes can also be described as being other shapes; e.g. a square is also a rhombus;
• recognise and understand right angles, parallel lines, lines of symmetry;
• develop their geometric reasoning skills.

Coordinate Sheets

Here is our collection of printable coordinate plane grids and coordinate worksheets.

Using these fun coordinate sheets is a great way to learn math in an enjoyable way.

• plot and write coordinates.
• Coordinate Plane Grid templates
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Have a look at these online symmetry games - a great way to learn symmetry and get instant feedback!

• Softschools Symmetry Game
• Sheppard Software Symmetry Activities

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