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

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?

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

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

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.

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

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

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?

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

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

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

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