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?
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?
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
A follow-up activity to Tiles in the Garden.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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. . . .
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!
What fractions of the largest circle are the two shaded regions?
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.
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?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
This article, written for teachers, discusses the merits of
different kinds of resources: those which involve exploration and
those which centre on calculation.
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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).
Determine the total shaded area of the 'kissing triangles'.
A task which depends on members of the group noticing the needs of
others and responding.
Derive a formula for finding the area of any kite.
Can you work out the area of the inner square and give an
explanation of how you did it?
What happens to the area and volume of 2D and 3D shapes when you
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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
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?
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.
How would you move the bands on the pegboard to alter these shapes?
How many tiles do we need to tile these patios?
Can you find the area of a parallelogram defined by two vectors?
An investigation that gives you the opportunity to make and justify
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
This article for teachers gives some food for thought when teaching
ideas about area.
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
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
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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.
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
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. . . .
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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. . . .
Here are many ideas for you to investigate - all linked with the