Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

How many different triangles can you make on a circular pegboard that has nine pegs?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

Can you find all the different triangles on these peg boards, and find their angles?

How would you move the bands on the pegboard to alter these shapes?

Board Block game for two. Can you stop your partner from being able to make a shape on the board?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Choose the size of your pegboard and the shapes you can make. Can you work out the strategies needed to block your opponent?

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 find triangles on a 9-point circle? Can you work out their angles?

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

The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .

This article for teachers explains why geoboards are such an invaluable resource and introduces several tasks which make use of them.

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.