Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

In the four examples below 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".

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

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

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

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

Various solids are lowered into a beaker of water. How does the water level rise in each case?

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

Where should runners start the 200m race so that they have all run the same distance by the finish?

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

A collection of short Stage 4 problems on area and volume.