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

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

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

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?

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

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

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).

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!

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

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.

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?

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.

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?

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

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

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

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 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.

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.

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

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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.

A follow-up activity to Tiles in the Garden.

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?

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

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

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.

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

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

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.

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. . . .

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

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

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.

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

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?

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.

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.

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?

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