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

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. . . .

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

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

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

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

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

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

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. . . .

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

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.

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

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?

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?

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

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

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

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

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.

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).

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?

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?

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

A follow-up activity to Tiles in the Garden.

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!

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

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

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?

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. . . .

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.

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

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?

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 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.

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

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

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.

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?

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.

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

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?

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.