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.

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 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. . . .

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

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

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?

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

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

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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

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.

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

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

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

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

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?

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.

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!

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.

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

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. . . .

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?

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

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

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. . . .

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

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

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?

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

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

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

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 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.

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

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?

A follow-up activity to Tiles in the Garden.

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. . . .

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

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

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?

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

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

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

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

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

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

What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.