Perimeter, area and volume - Stage 4

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?

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?

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

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

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

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 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 ?