There are 64 results
Broad Topics > Angles, Polygons, and Geometrical Proof > 2D shapes and their properties
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?
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.
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?
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?
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.
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. . . .
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?
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.
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?
LOGO Challenge 12 - Concentric Circles
Age 11 to 16
Challenge Level
Can you reproduce the design comprising a series of concentric circles? Test your understanding of the realtionship betwwn the circumference and diameter of a circle.
LOGO Challenge 11 - More on Circles
Age 11 to 16
Challenge Level
Thinking of circles as polygons with an infinite number of sides - but how does this help us with our understanding of the circumference of circle as pi x d? This challenge investigates. . . .
First Forward Into Logo 4: Circles
Age 7 to 16
Challenge Level
Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?
Pentagonal
Age 14 to 16
Challenge Level
Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)? Can you prove it to be true for a rectangle or a hexagon?
Circles, Circles Everywhere
Age 7 to 14
This article for pupils gives some examples of how circles have featured in people's lives for centuries.
From One Shape to Another
Age 7 to 14
Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.
Pi, a Very Special Number
Age 7 to 14
Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.
Circumnavigation
Age 14 to 16
Challenge Level
The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
Bow Tie
Age 11 to 14
Challenge Level
Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.
LOGO Challenge 8 - Rhombi
Age 7 to 16
Challenge Level
Explore patterns based on a rhombus. How can you enlarge the pattern - or explode it?
LOGO Challenge 2 - Diamonds Are Forever
Age 7 to 16
Challenge Level
The challenge is to produce elegant solutions. Elegance here implies simplicity. The focus is on rhombi, in particular those formed by jointing two equilateral triangles along an edge.
LOGO Challenge 6 - Triangles and Stars
Age 11 to 16
Challenge Level
Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.
LOGO Challenge - Circles as Animals
Age 11 to 16
Challenge Level
See if you can anticipate successive 'generations' of the two animals shown here.
LOGO Challenge 10 - Circles
Age 11 to 16
Challenge Level
In LOGO circles can be described in terms of polygons with an infinite (in this case large number) of sides - investigate this definition further.
Arclets Explained
Age 11 to 16
This article gives an wonderful insight into students working on the Arclets problem that first appeared in the Sept 2002 edition of the NRICH website.
Squaring the Circle and Circling the Square
Age 14 to 16
Challenge Level
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Playground Snapshot
Age 7 to 14
Challenge Level
The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?
Squaring the Circle
Age 11 to 14
Challenge Level
Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .
Semi-square
Age 14 to 16
Challenge Level
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
Crescents and Triangles
Age 14 to 16
Challenge Level
Can you find a relationship between the area of the crescents and the area of the triangle?
Pent
Age 14 to 18
Challenge Level
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
Tricircle
Age 14 to 16
Challenge Level
The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .
Rhombus in Rectangle
Age 14 to 16
Challenge Level
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
2001 Spatial Oddity
Age 11 to 14
Challenge Level
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
The Medieval Octagon
Age 14 to 16
Challenge Level
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
Holly
Age 14 to 16
Challenge Level
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.
Lying and Cheating
Age 11 to 14
Challenge Level
Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!
Three Four Five
Age 14 to 16
Challenge Level
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
NRICH is part of the family of activities in the Millennium Mathematics Project.