A follow-up activity to Tiles in the Garden.

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

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

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. . . .

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 fractions of the largest circle are the two shaded regions?

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?

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

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?

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

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.

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

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

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. . . .

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?

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

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

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

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

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

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

A simple visual exploration into halving and doubling.

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

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

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!

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

This article for teachers gives some food for thought when teaching ideas about area.

Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.

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

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

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?

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.

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. . . .

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?

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

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?

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.

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

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. . . .

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

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

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

An investigation that gives you the opportunity to make and justify predictions.

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?

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