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?

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

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. . . .

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

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.

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

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

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

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

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

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.

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?

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

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

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.

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

Prove that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.

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?

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

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. . . .

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

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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

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.

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

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

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

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.

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.

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

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.

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.

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!

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?

A follow-up activity to Tiles in the Garden.

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

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

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?

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. . . .

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?

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