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.

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 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. . . .

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?

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.

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

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!

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

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

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

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

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?

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. . . .

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

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 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.

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

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

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

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.

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. . . .

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?

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.

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

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

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.

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?

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.

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. . . .

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.

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

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

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?

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

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

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. . . .

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. . . .

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

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

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

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.

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.

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?