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 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?
Explore one of these five pictures.
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?
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
A follow-up activity to Tiles in the Garden.
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.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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. . . .
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.
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.
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?
What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?
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.
What's special about the area of quadrilaterals drawn in a square?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
This article for teachers gives some food for thought when teaching ideas about area.
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!
Look at the mathematics that is all around us - this circular window is a wonderful example.
We started drawing some quadrilaterals - can you complete them?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
Determine the total shaded area of the 'kissing triangles'.
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.
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?
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. . . .
A task which depends on members of the group noticing the needs of others and responding.
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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. . . .
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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).
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. . . .
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.
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 fractions of the largest circle are the two shaded regions?
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.
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?
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?
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 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?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?