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?

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?

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?

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

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. . . .

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.

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

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.

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

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. . . .

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

Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.

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.

A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?

Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area

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.

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?

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

Make an eight by eight square, the layout is the same as a chessboard. You can print out and use the square below. What is the area of the square? Divide the square in the way shown by the red dashed. . . .

What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

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

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?

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

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

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

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. . . .

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.

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

Can you work out the area of the inner square and give an explanation of how you did it?

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. . . .

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.

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

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

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

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?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

A follow-up activity to Tiles in the Garden.

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

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!

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

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.

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?