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.

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?

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

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?

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. . . .

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

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?

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

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.

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

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

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.

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

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

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

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?

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

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

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

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?

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?

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

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?

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

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

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?

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

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

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

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?

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.

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!

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

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

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.

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. . . .

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?

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?

A follow-up activity to Tiles in the Garden.

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

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.

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. . . .