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?

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.

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

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

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.

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

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

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 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).

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

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.

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

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

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

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 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?

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

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.

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

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

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

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.

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. . . .

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 fractions of the largest circle are the two shaded regions?

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

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

A follow-up activity to Tiles in the Garden.

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.

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

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.

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. . . .

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

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

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. . . .

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?

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!

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.

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.

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 small circle in a square in a big circle in a trapezium. Using the measurements and clue given, find the area of the trapezium.

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.