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. . . .

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

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

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?

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?

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.

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?

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.

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

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

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

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?

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?

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

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

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.

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

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

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.

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. . . .

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. . . .

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

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.

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?

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?

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

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?

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 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. . . .

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

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?

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?

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

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. . . .

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

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.

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

A trapezium is divided into four triangles by its diagonals. Suppose the two triangles containing the parallel sides have areas a and b, what is the area of the trapezium?

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

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

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

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

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

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.