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

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.

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?

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).

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. . . .

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

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?

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

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?

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

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

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

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!

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. . . .

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

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.

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

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. . . .

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?

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. . . .

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. . . .

You have pitched your tent (the red triangle) on an island. Can you move it to the position shown by the purple triangle making sure you obey the rules?

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

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?

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.

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?

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

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?

Look at the mathematics that is all around us - this circular window is a wonderful example.

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.

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

How would you move the bands on the pegboard to alter these shapes?

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.

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.

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. . . .

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

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. . . .

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?

Have a good look at these images. Can you describe what is happening? There are plenty more images like this on NRICH's Exploring Squares CD.

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?

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

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.