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?

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

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.

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.

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. . . .

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

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

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 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).

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

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

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

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?

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

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?

Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area

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

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

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

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

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.

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.

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?

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

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

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

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. . . .

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

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!

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.

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?

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.

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

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?

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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?

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

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

A follow-up activity to Tiles in the Garden.

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .

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?

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

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

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. . . .

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. . . .

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

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