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

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

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. . . .

How many centimetres of rope will I need to make another mat just like the one I have here?

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.

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. . . .

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. . . .

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?

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. . . .

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.

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.

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. . . .

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

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?

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?

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

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

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

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?

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. . . .

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

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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

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.

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

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).

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

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

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

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

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

Use the information on these cards to draw the shape that is being described.

A simple visual exploration into halving and doubling.

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

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

Are these statements always true, sometimes true or never true?

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

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

A follow-up activity to Tiles in the Garden.

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