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?

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.

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

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

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

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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. . . .

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.

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

A follow-up activity to Tiles in the Garden.

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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

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

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

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?

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

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?

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

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?

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

Determine the total shaded area of the 'kissing 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?

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

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

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?

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

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

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

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

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?

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

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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

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.

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

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?

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

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

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

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.

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

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?

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. . . .