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There are **70** NRICH Mathematical resources connected to **Symmetry**, you may find related items under Transformations and constructions.

Problem
Primary curriculum
Secondary curriculum
### Always, Sometimes or Never? Shape

Are these statements always true, sometimes true or never true?

Age 7 to 11

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Problem
Primary curriculum
Secondary curriculum
### Shape Draw

Use the information on these cards to draw the shape that is being described.

Age 7 to 11

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Problem
Primary curriculum
Secondary curriculum
### National Flags

This problem explores the shapes and symmetries in some national flags.

Age 7 to 11

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Primary curriculum
Secondary curriculum
### Poly Plug Pattern

Create a pattern on the small grid. How could you extend your pattern on the larger grid?

Age 5 to 7

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Primary curriculum
Secondary curriculum
### Exploded Squares

This practical activity challenges you to create symmetrical designs by cutting a square into strips.

Age 5 to 7

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Primary curriculum
Secondary curriculum
### Attractive Rotations

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Age 11 to 14

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Primary curriculum
Secondary curriculum
### Reflector ! Rotcelfer

Can you place the blocks so that you see the reflection in the picture?

Age 7 to 11

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Primary curriculum
Secondary curriculum
### Coordinate Challenge

Use the clues about the symmetrical properties of these letters to place them on the grid.

Age 7 to 11

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Primary curriculum
Secondary curriculum
### Stringy Quads

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

Age 7 to 11

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Problem
Primary curriculum
Secondary curriculum
### Octa-flower

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Symmetry Challenge

How many symmetric designs can you make on this grid? Can you find them all?

Age 7 to 11

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Primary curriculum
Secondary curriculum
### Shady Symmetry

How many different symmetrical shapes can you make by shading triangles or squares?

Age 11 to 14

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Primary curriculum
Secondary curriculum
### Reflecting Squarely

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Age 11 to 14

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Primary curriculum
Secondary curriculum
### A Cartesian Puzzle

Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.

Age 7 to 11

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Primary curriculum
Secondary curriculum
### Attractive Tablecloths

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Arclets

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Witch of Agnesi

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Folium of Descartes

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Logosquares

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Mean Geometrically

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Colouring Triangles

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

Age 5 to 7

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Primary curriculum
Secondary curriculum
### A Roll of Patterned Paper

A design is repeated endlessly along a line - rather like a stream of paper coming off a roll. Make a strip that matches itself after rotation, or after reflection

Age 14 to 16

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Problem
Primary curriculum
Secondary curriculum
### Symmetric Trace

Points off a rolling wheel make traces. What makes those traces have symmetry?

Age 14 to 16

Challenge Level

General
Primary curriculum
Secondary curriculum
### Building Patterns

Can you deduce the pattern that has been used to lay out these bottle tops?

Age 7 to 11

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Emmy Noether

Find out about Emmy Noether, whose ideas linked physics and algebra, and whom Einstein described as a 'creative mathematical genius'.

Age 7 to 14

General
Primary curriculum
Secondary curriculum
### Watch Those Wheels

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

Age 5 to 7

Challenge Level

General
Primary curriculum
Secondary curriculum
### Lafayette

What mathematical words can be used to describe this floor covering? How many different shapes can you see inside this photograph?

Age 7 to 11

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Primary curriculum
Secondary curriculum
### Pitchfork

Plot the graph of x^y = y^x in the first quadrant and explain its properties.

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Sliced

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Symmetrical Semaphore

Someone at the top of a hill sends a message in semaphore to a friend in the valley. A person in the valley behind also sees the same message. What is it?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Hidden Meaning

What is the missing symbol? Can you decode this in a similar way?

Age 7 to 11

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Pattern Power

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

Age 5 to 14

Problem
Primary curriculum
Secondary curriculum
### Two Triangles in a Square

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Friezes

Some local pupils lost a geometric opportunity recently as they surveyed the cars in the car park. Did you know that car tyres, and the wheels that they on, are a rich source of geometry?

Age 11 to 14

Article
Primary curriculum
Secondary curriculum
### Tournament Scheduling

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

Age 11 to 16

Problem
Primary curriculum
Secondary curriculum
### Classifying Solids Using Angle Deficiency

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Age 11 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### The Frieze Tree

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Age 11 to 16

Article
Primary curriculum
Secondary curriculum
### Frieze Patterns in Cast Iron

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

Age 11 to 16

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Primary curriculum
Secondary curriculum
### Overlap

A red square and a blue square overlap. Is the area of the overlap always the same?

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Square Pizza

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Prime Magic

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

Age 7 to 16

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Primary curriculum
Secondary curriculum
### Rhombicubocts

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices does each solid have?

Age 11 to 14

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Primary curriculum
Secondary curriculum
### Encircling

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Holly

The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Eight Dominoes

Using the 8 dominoes make a square where each of the columns and rows adds up to 8

Age 7 to 16

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Primary curriculum
Secondary curriculum
### A Problem of Time

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

Age 14 to 16

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Primary curriculum
Secondary curriculum
### Maltese Cross

Sketch the graph of $xy(x^2 - y^2) = x^2 + y^2$ consisting of four curves and a single point at the origin. Convert to polar form. Describe the symmetries of the graph.

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Cocked Hat

Sketch the graphs for this implicitly defined family of functions.

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Cut Cube

Find the shape and symmetries of the two pieces of this cut cube.

Age 16 to 18

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Primary curriculum
Secondary curriculum
### Polydron

This activity investigates how you might make squares and pentominoes from Polydron.

Age 7 to 11

Challenge Level