You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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.

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

There are many different methods to solve this geometrical problem - how many can you find?

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

If you move the tiles around, can you make squares with different coloured edges?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

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?

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

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

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

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?

Explore the relationships between different paper sizes.

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?