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?

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

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

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.

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

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?

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

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

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

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.

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?

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

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

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?

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?

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. . . .

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

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

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?

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

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

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.

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.

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

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

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 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?

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?

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

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.

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

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.

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.

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!

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

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.

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

This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.

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?

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

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.

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?

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. . . .