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?

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. . . .

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.

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. . . .

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 fractions of the largest circle are the two shaded regions?

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

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?

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

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

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

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

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

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

A follow-up activity to Tiles in the Garden.

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

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?

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?

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

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

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

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. . . .

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

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?

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

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

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?

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

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. . . .

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.

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

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!

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?

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

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

It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?

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?

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?

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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. . . .

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

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. . . .

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. . . .