Square It
Stage: 3 and 4 Challenge Level: 
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Square Coordinates
Stage: 3 Challenge Level:
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
Triangles in Circles
Stage: 3 Challenge Level:
Can you find triangles on a 9-point circle? Can you work out their angles?
Subtended Angles
Stage: 3 Challenge Level:
What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?
Right Angles
Stage: 3 Challenge Level:
Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
Quadrilaterals Game
Stage: 3 Challenge Level:
A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and two properties of the shape.
Shapely Pairs
Stage: 3 Challenge Level:
A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...
Property Chart
Stage: 3 Challenge Level:
A game in which players take it in turns to try to draw quadrilaterals (or triangles) with particular properties. Is it possible to fill the game grid?
Semi-regular Tessellations
Stage: 3 Challenge Level:
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
Cyclic Quadrilaterals
Stage: 3 and 4 Challenge Level:
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Which Solids Can We Make?
Stage: 3 Challenge Level:
Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?
Opposite Vertices
Stage: 3 Challenge Level:
Can you recreate squares and rhombuses if you are only given a side or a diagonal?
Guess my Quad
Stage: 3 Challenge Level:
How many questions do you need to identify my quadrilateral?
Completing Quadrilaterals
Stage: 3 Challenge Level:
We started drawing some quadrilaterals - can you complete them?
Of All the Areas
Stage: 4 Challenge Level:
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Tilted Squares
Stage: 3 Challenge Level:
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Quadrilaterals in a Square
Stage: 3 Challenge Level:
What's special about the area of quadrilaterals drawn in a square?
Triangle in the Corner
Stage: 3 Short Challenge Level:
A triangle is shaded within a regular hexagon. Can you find its area?
Perimeter in a Hexagon
Stage: 3 Short Challenge Level:
Can you find the perimeter of this triangle inscribed in a hexagon?
Tricky Tessellations
Stage: 3 Short Challenge Level:
Can you work out the fraction of the tiles that are painted black in this pattern?
Descending Angles
Stage: 3 Short Challenge Level:
Given four of the angles in two triangles, can you find the smallest angle overall?
Clock Face Angles
Stage: 3 Short Challenge Level:
The time is 20:14. What is the smaller angle between the hour hand and the minute hand on an accurate analogue clock?
Overlapping Beer Mats
Stage: 3 Short Challenge Level:
Can you find the area of the overlap when these two beer mats are placed on top of each other?
Clock Angle
Stage: 3 Short Challenge Level:
What is the angle between the the hands of a clock at 8:24?
Two Triangles
Stage: 3 Short Challenge Level:
Prove that the angle marked $a$ is half the size of the angle marked $b$.
Inner Rectangle
Stage: 3 Short Challenge Level:
If the shape on the inside is a rectangle, what can you say about the shape on the outside?
No Rights
Stage: 3 Short Challenge Level:
Prove that the angle bisectors of a triangle can never meet at right angles.
Perimeter Puzzle
Stage: 3 Short Challenge Level:
If four copies of this triangle are joined together to form a parallelogram, what is the largest possible perimeter of the parallelogram?
Diagonal Side
Stage: 3 Short Challenge Level:
Can you work out the area of a square drawn on a diagonal?
Add All the Angles
Stage: 3 Short Challenge Level:
Find the sum of all of the angles denoted by letters in this diagram
NRICH is part of the family of activities in the Millennium Mathematics Project.