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

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

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. . . .

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.

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

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

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?

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

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 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?

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

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

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

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

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

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

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

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

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

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.

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.

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?

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

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

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

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.

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

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 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?

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

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 follow-up activity to Tiles in the Garden.

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!

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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

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?

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

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?

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.

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

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 find the areas of the trapezia in this sequence?