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

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

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

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

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.

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

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. . . .

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. . . .

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?

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

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 ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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

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?

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

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.

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

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 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?

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?

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

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. . . .

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?

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.

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

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

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

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

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).

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. . . .

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

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.

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

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

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!

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.

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?

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 follow-up activity to Tiles in the Garden.

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.

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