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

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. . . .

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).

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

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. . . .

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 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?

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.

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

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

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

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

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

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. . . .

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?

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

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!

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.

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?

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

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares?

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

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?

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.

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

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

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?

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

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

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

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

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

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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?

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. . . .

You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.

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

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

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

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

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?