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

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

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

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

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

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

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

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

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?

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

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!

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

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?

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?

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?

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.

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

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 draw a square in which the perimeter is numerically equal to the area?

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.

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

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

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?

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?

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?

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?

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

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?

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?

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

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.

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

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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.

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?

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).

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

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. . . .

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.

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

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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?