The graph below is an oblique coordinate system based on 60 degree angles. It was drawn on isometric paper. What kinds of triangles do these points form?

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

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

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Make a flower design using the same shape made out of different sizes of paper.

The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the shapes?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

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

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Make an equilateral triangle by folding paper and use it to make patterns of your own.

This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.

Can you sketch triangles that fit in the cells in this grid? Which ones are impossible? How do you know?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

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?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

You have pitched your tent (the red triangle) on an island. Can you move it to the position shown by the purple triangle making sure you obey the rules?

This interactivity allows you to sort logic blocks by dragging their images.

What do these two triangles have in common? How are they related?

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

Determine the total shaded area of the 'kissing triangles'.

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

Have you noticed that triangles are used in manmade structures? Perhaps there is a good reason for this? 'Test a Triangle' and see how rigid triangles are.

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

The triangles in these sets are similar - can you work out the lengths of the sides which have question marks?

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.

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 each work out what shape you have part of on your card? What will the rest of it look like?

Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

I cut this square into two different shapes. What can you say about the relationship between them?

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

Jennifer Piggott and Charlie Gilderdale describe a free interactive circular geoboard environment that can lead learners to pose mathematical questions.

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?