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?

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

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. . . .

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

A small circle in a square in a big circle in a trapezium. Using the measurements and clue given, find the area of the trapezium.

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

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

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

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

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

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

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

A hallway floor is tiled and each tile is one foot square. Given that the number of tiles around the perimeter is EXACTLY half the total number of tiles, find the possible dimensions of the hallway.

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

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

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .

A trapezium is divided into four triangles by its diagonals. Suppose the two triangles containing the parallel sides have areas a and b, what is the area of the trapezium?

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

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?

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

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?

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

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

A follow-up activity to Tiles in the Garden.

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. . . .

Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?

Can you find the areas of the trapezia in this sequence?

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

Determine the total shaded area of the 'kissing triangles'.

This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

A task which depends on members of the group noticing the needs of others and responding.

What happens to the area and volume of 2D and 3D shapes when you enlarge them?

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.