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. . . .

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

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

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .

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

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

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

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

What fractions of the largest circle are the two shaded regions?

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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

Can you maximise the area available to a grazing goat?

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

Can you find a relationship between the area of the crescents and the area of the triangle?

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

Find the perimeter and area of a holly leaf that will not lie flat (it has negative curvature with 'circles' having circumference greater than 2πr).

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?

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and. . . .

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

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?

If I print this page which shape will require the more yellow ink?

Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.