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

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

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?

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

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

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

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

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

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!

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. . . .

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.

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

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

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

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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?

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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

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

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?

A follow-up activity to Tiles in the Garden.

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

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?

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.

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

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?

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

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.

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

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?

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.

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

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 circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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?

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

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

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.

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

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.

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.