There are 70 NRICH Mathematical resources connected to Symmetry, you may find related items under Transformations and constructions.
Broad Topics > Transformations and constructions > Symmetry
Always, Sometimes or Never? Shape
Age 7 to 11 Challenge Level:
Are these statements always true, sometimes true or never true?
Shape Draw
Age 7 to 11 Challenge Level:
Use the information on these cards to draw the shape that is being described.
National Flags
Age 7 to 11 Challenge Level:
This problem explores the shapes and symmetries in some national flags.
Poly Plug Pattern
Age 5 to 7 Challenge Level:
Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?
Exploded Squares
Age 5 to 7 Challenge Level:
This practical activity challenges you to create symmetrical designs by cutting a square into strips.
Attractive Rotations
Age 11 to 14 Challenge Level:
Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...
Coordinate Challenge
Age 7 to 11 Challenge Level:
Use the clues about the symmetrical properties of these letters to place them on the grid.
Stringy Quads
Age 7 to 11 Challenge Level:
This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!
Octa-flower
Age 16 to 18 Challenge Level:
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?
Symmetry Challenge
Age 7 to 11 Challenge Level:
Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.
Shady Symmetry
Age 11 to 14 Challenge Level:
How many different symmetrical shapes can you make by shading triangles or squares?
Reflecting Squarely
Age 11 to 14 Challenge Level:
In how many ways can you fit all three pieces together to make shapes with line symmetry?
A Cartesian Puzzle
Age 7 to 11 Challenge Level:
Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.
Attractive Tablecloths
Age 14 to 16 Challenge Level:
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?
Arclets
Age 14 to 16 Challenge Level:
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".
Witch of Agnesi
Age 16 to 18 Challenge Level:
Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.
Folium of Descartes
Age 16 to 18 Challenge Level:
Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.
Logosquares
Age 16 to 18 Challenge Level:
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.
Mean Geometrically
Age 16 to 18 Challenge Level:
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?
Colouring Triangles
Age 5 to 7 Challenge Level:
Explore ways of colouring this set of triangles. Can you make symmetrical patterns?
Gibraltar Geometry
Age 11 to 14 Challenge Level:
Take a look at the photos of tiles at a school in Gibraltar. What questions can you ask about them?
Flower Power
Age 11 to 16 Challenge Level:
Create a symmetrical fabric design based on a flower motif - and realise it in Logo.
More Dicey Decisions
Age 16 to 18 Challenge Level:
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Dicey Decisions
Age 16 to 18 Challenge Level:
Can you devise a fair scoring system when dice land edge-up or corner-up?
Troublesome Dice
Age 11 to 14 Challenge Level:
When dice land edge-up, we usually roll again. But what if we didn't...?
Paint Rollers for Frieze Patterns.
Age 11 to 16
Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.
Reflector ! Rotcelfer
Age 7 to 11 Challenge Level:
Can you place the blocks so that you see the reflection in the picture?
Topkapi Palace
Age 7 to 14 Challenge Level:
These images are taken from the Topkapi Palace in Istanbul, Turkey. Can you work out the basic unit that makes up each pattern? Can you continue the pattern? Can you see any similarities and. . . .
Drawing Celtic Knots
Age 11 to 14 Challenge Level:
Here is a chance to create some Celtic knots and explore the mathematics behind them.
Trominoes
Age 11 to 16 Challenge Level:
Can all but one square of an 8 by 8 Chessboard be covered by Trominoes?
Tiles in a Public Building
Age 7 to 11 Challenge Level:
What is the same and what is different about these tiling patterns and how do they contribute to the floor as a whole?
Animated Triangles
Age 5 to 7 Challenge Level:
Watch this "Notes on a Triangle" film. Can you recreate parts of the film using cut-out triangles?
Triangle Shapes
Age 5 to 11 Challenge Level:
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Turning Tangles
Age 11 to 14 Challenge Level:
Look carefully at the video of a tangle and explain what's happening.
Dancing with Maths
Age 7 to 16
An article for students and teachers on symmetry and square dancing. What do the symmetries of the square have to do with a dos-e-dos or a swing? Find out more?
Making Maths: Five-point Snowflake
Age 7 to 11 Challenge Level:
Follow these instructions to make a five-pointed snowflake from a square of paper.
Making Maths: Indian Window Screen
Age 7 to 11 Challenge Level:
Can you recreate this Indian screen pattern? Can you make up similar patterns of your own?
Making Maths: Snowflakes
Age 7 to 11 Challenge Level:
It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!
Rotations Are Not Single Round Here
Age 14 to 16 Challenge Level:
I noticed this about streamers that have rotation symmetry : if there was one centre of rotation there always seems to be a second centre that also worked. Can you find a design that has only. . . .
One Reflection Implies Another
Age 14 to 16 Challenge Level:
When a strip has vertical symmetry there always seems to be a second place where a mirror line could go. Perhaps you can find a design that has only one mirror line across it. Or, if you thought that. . . .
A Roll of Patterned Paper
Age 14 to 16 Challenge Level:
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
Symmetric Trace
Age 14 to 16 Challenge Level:
Points off a rolling wheel make traces. What makes those traces have symmetry?
Building Patterns
Age 7 to 11 Challenge Level:
Can you deduce the pattern that has been used to lay out these bottle tops?
Watch Those Wheels
Age 5 to 7 Challenge Level:
Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!
Lafayette
Age 7 to 11 Challenge Level:
What mathematical words can be used to describe this floor covering? How many different shapes can you see inside this photograph?
Pitchfork
Age 16 to 18 Challenge Level:
Plot the graph of x^y = y^x in the first quadrant and explain its properties.
Sliced
Age 14 to 16 Challenge Level:
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?
Symmetrical Semaphore
Age 7 to 11 Challenge Level:
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?
