Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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
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.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
A task which depends on members of the group noticing the needs of
others and responding.
What fractions of the largest circle are the two shaded regions?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
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!
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?
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.
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. . . .
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?
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?
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
Can you work out the area of the inner square and give an
explanation of how you did it?
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. . . .
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. . . .
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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).
Derive a formula for finding the area of any kite.
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.
These practical challenges are all about making a 'tray' and covering it with paper.
This article for teachers gives some food for thought when teaching
ideas about area.
A simple visual exploration into halving and doubling.
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.
Can you maximise the area available to a grazing goat?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
An investigation that gives you the opportunity to make and justify
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 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 can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
What do these two triangles have in common? How are they related?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
How many tiles do we need to tile these patios?
How would you move the bands on the pegboard to alter these shapes?