Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

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.

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?

How many different triangles can you make which consist of the centre point and two of the points on the edge? Can you work out each of their angles?

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?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

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.

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...

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 combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

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?

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

How many questions do you need to identify my quadrilateral?

We started drawing some quadrilaterals - can you complete them?

Draw some stars and measure the angles at their points. Can you find and prove a result about their sum?

Can you work out how these polygon pictures were drawn, and use that to figure out their angles?

Take an equilateral triangle and cut it into smaller pieces. What can you do with them?

A collection of short Stage 3 problems on Angles and Polygons.

Given four of the angles in two triangles, can you find the smallest angle overall?

The time is 20:14. What is the smaller angle between the hour hand and the minute hand on an accurate analogue clock?

Can you find the area of the overlap when these two beer mats are placed on top of each other?

What is the angle between the the hands of a clock at 8:24?

Prove that the angle marked $a$ is half the size of the angle marked $b$.

If the shape on the inside is a rectangle, what can you say about the shape on the outside?

Prove that the angle bisectors of a triangle can never meet at right angles.