Area - circles, sectors and segments

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

Can you maximise the area available to a grazing goat?

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?

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

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?

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

What is the same and what is different about these circle questions? What connections can you make?

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?