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?

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

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?

Can you find the shortest distance between the semicircles given the area between them?

What is the ratio of the areas of the squares in the diagram?

Two vases are cylindrical in shape. Can you work out the original depth of the water in the larger vase?

Can you locate the point on an annulus that splits it into two areas?

A solid metal cone is melted down and turned into spheres. How many spheres can be made?

Cutting a rectangle from a corner to a point on the opposite side splits its area in the ratio 1:2. What is the ratio of a:b?

Can you find the area of the yellow part of this snake's eye?

When the roll of toilet paper is half as wide, what percentage of the paper is left?