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

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

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

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.

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?

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. . . .

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

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.

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

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?

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?

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?

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.

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

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.

Make an eight by eight square, the layout is the same as a chessboard. You can print out and use the square below. What is the area of the square? Divide the square in the way shown by the red dashed. . . .

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

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

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

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

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

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?

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

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 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?

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

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

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.

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?

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.

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?

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?

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).

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

You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.

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

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

A follow-up activity to Tiles in the Garden.

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. . . .

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. . . .

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

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

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?

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

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?