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

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 area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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. . . .

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.

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

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

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

A follow-up activity to Tiles in the Garden.

Analyse these beautiful biological images and attempt to rank them in size order.

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

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

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

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

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

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?

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

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!

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?

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.

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.

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

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?

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.

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

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

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.

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?

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

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?

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. . . .

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

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?

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

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

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?

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. . . .

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

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

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