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

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

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. . . .

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

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?

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.

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. . . .

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.

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?

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. . . .

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. . . .

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

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. . . .

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?

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

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

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?

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

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.

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

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

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?

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.

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.

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

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?

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

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.

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

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

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

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

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?

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

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?

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

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. . . .

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?

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?

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

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

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

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.