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?

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

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

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

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. . . .

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

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 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.

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. . . .

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. . . .

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

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. . . .

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

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.

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?

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

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

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?

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

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?

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.

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.

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

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

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?

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?

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?

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!

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.

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

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.

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

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?

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

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?

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?