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?
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
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?
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).
I cut this square into two different shapes. What can you say about the relationship between them?
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 fractions of the largest circle are the two shaded regions?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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. . . .
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Determine the total shaded area of the 'kissing triangles'.
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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Grandpa was measuring a rug using yards, feet and inches. Can you help William to work out its area?
Measure problems at primary level that may require determination.
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!
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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.
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.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
How would you move the bands on the pegboard to alter these shapes?
Are these statements always true, sometimes true or never true?
Measure problems at primary level that require careful consideration.
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.
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.
Measure problems for inquiring primary learners.
Use the information on these cards to draw the shape that is being described.
A task which depends on members of the group noticing the needs of others and responding.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
A simple visual exploration into halving and doubling.
How many tiles do we need to tile these patios?
Explore one of these five pictures.
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?
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
Look at the mathematics that is all around us - this circular window is a wonderful example.
A follow-up activity to Tiles in the Garden.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
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?
Measure problems for primary learners to work on with others.
How many centimetres of rope will I need to make another mat just like the one I have here?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
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.
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.
Can you work out the area of the inner square and give an explanation of how you did it?
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?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?