Area - circles, sectors and segments

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

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

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?

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

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

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

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.

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?