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

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?

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

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.

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

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 follow-up activity to Tiles in the Garden.

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

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. . . .

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

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. . . .

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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!

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

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

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

Explore this interactivity and see if you can work out what it does. Could you use it to estimate the area of a shape?

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

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 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?

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

What do these two triangles have in common? How are they related?

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?

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

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

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

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

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. . . .

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

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

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.

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. . . .

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

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

These practical challenges are all about making a 'tray' and covering it with paper.

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.

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. . . .

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. . . .

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. . . .

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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 largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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