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

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

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

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

A follow-up activity to Tiles in the Garden.

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

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

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

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.

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?

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

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

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?

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

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

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?

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.

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

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.

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?

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?

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

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?

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

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

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?

Can you find the areas of the trapezia in this sequence?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

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

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?

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!

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.

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?

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?

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.

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

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. . . .

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?