There are 116 NRICH Mathematical resources connected to 2D shapes and their properties, you may find related items under Angles, Polygons, and Geometrical Proof.
Broad Topics > Angles, Polygons, and Geometrical Proof > 2D shapes and their properties
Always, Sometimes or Never? KS1
Age 5 to 7 Challenge Level:
Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?
What's Happening?
Age 5 to 7 Challenge Level:
Shapes are added to other shapes. Can you see what is happening? What is the rule?
Olympic Rings
Age 5 to 7 Challenge Level:
This problem is intended to get children to look really hard at something they will see many times in the next few months.
Poly Plug Rectangles
Age 5 to 14 Challenge Level:
The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?
Track Design
Age 14 to 16 Challenge Level:
Where should runners start the 200m race so that they have all run the same distance by the finish?
Shaping It
Age 5 to 11 Challenge Level:
These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.
Sorting Logic Blocks
Age 5 to 11 Challenge Level:
This activity focuses on similarities and differences between shapes.
Shapely Lines
Age 5 to 7 Challenge Level:
This challenge invites you to create your own picture using just straight lines. Can you identify shapes with the same number of sides and decorate them in the same way?
What Shape?
Age 7 to 14 Challenge Level:
This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.
Jig Shapes
Age 5 to 11 Challenge Level:
Can you each work out what shape you have part of on your card? What will the rest of it look like?
Curvy Areas
Age 14 to 16 Challenge Level:
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Overlapping Again
Age 7 to 11 Challenge Level:
What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.
2 Rings
Age 5 to 7 Challenge Level:
The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?
Torn Shapes
Age 7 to 11 Challenge Level:
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
Trapezium Four
Age 14 to 16 Challenge Level:
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
Spokes
Age 16 to 18 Challenge Level:
Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.
Board Block Challenge
Age 7 to 11 Challenge Level:
Choose the size of your pegboard and the shapes you can make. Can you work out the strategies needed to block your opponent?
Efficient Cutting
Age 11 to 14 Challenge Level:
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
Salinon
Age 14 to 16 Challenge Level:
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Rolling Around
Age 11 to 14 Challenge Level:
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
Semi-detached
Age 14 to 16 Challenge Level:
A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
Where Are They?
Age 7 to 11 Challenge Level:
Use the isometric grid paper to find the different polygons.
Approximating Pi
Age 14 to 18 Challenge Level:
By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?
Blue and White
Age 11 to 14 Challenge Level:
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Hex
Age 11 to 14 Challenge Level:
Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.
Quadarc
Age 14 to 16 Challenge Level:
Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the. . . .
Orthogonal Circle
Age 16 to 18 Challenge Level:
Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.
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.
Baby Circle
Age 16 to 18 Challenge Level:
A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
Chain of Changes
Age 5 to 7 Challenge Level:
Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?
Possible Pairs
Age 7 to 11 Challenge Level:
In this game, you turn over two cards and try to draw a triangle which has both properties.
Name That Triangle!
Age 7 to 11 Challenge Level:
Can you sketch triangles that fit in the cells in this grid? Which ones are impossible? How do you know?
Triangle or No Triangle?
Age 5 to 7 Challenge Level:
Here is a selection of different shapes. Can you work out which ones are triangles, and why?
Regular Rings 2
Age 7 to 11 Challenge Level:
What shape is made when you fold using this crease pattern? Can you make a ring design?
Towering Trapeziums
Age 14 to 16 Challenge Level:
Can you find the areas of the trapezia in this sequence?
What Shape for Two
Age 7 to 14 Challenge Level:
'What Shape?' activity for adult and child. Can you ask good questions so you can work out which shape your partner has chosen?
Gym Bag
Age 11 to 16 Challenge Level:
Can Jo make a gym bag for her trainers from the piece of fabric she has?
Witch's Hat
Age 11 to 16 Challenge Level:
What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?
Curve Spotting
Age 5 to 7 Challenge Level:
Can you spot circles, spirals and other types of curves in these photos?
Opening Patterns
Age 5 to 7 Challenge Level:
Look at some of the patterns in the Olympic Opening ceremonies and see what shapes you can spot.
A Rational Search
Age 16 to 18 Challenge Level:
Investigate constructible images which contain rational areas.
Sports Equipment
Age 7 to 11 Challenge Level:
If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?
Opposite Vertices
Age 11 to 14 Challenge Level:
Can you recreate squares and rhombuses if you are only given a side or a diagonal?
Bicentric Quadrilaterals
Age 14 to 16 Challenge Level:
Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
Logic Block Collections
Age 5 to 7 Challenge Level:
What do you think is the same about these two Logic Blocks? What others do you think go with them in the set?
Shaping Up
Age 7 to 11 Challenge Level:
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
What's Inside/outside/under the Box?
Age 7 to 14
This article describes investigations that offer opportunities for children to think differently, and pose their own questions, about shapes.
Like a Circle in a Spiral
Age 7 to 16 Challenge Level:
A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?
Spirostars
Age 16 to 18 Challenge Level:
A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?
NRICH is part of the family of activities in the Millennium Mathematics Project.