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

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?

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

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

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

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

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

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

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.

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?

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. . . .

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.

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?

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

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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?

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

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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?

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. . . .

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.

Can you draw the height-time chart as this complicated vessel fills with water?

Analyse these beautiful biological images and attempt to rank them in size order.

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

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

A follow-up activity to Tiles in the Garden.

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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?

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?

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).

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

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

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

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.

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

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

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?

Can you find the area of a parallelogram defined by two vectors?

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

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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!

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.