Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
How many centimetres of rope will I need to make another mat just
like the one I have here?
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.
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?
A follow-up activity to Tiles in the Garden.
Can you work out the area of the inner square and give an
explanation of how you did it?
Explore one of these five pictures.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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.
A task which depends on members of the group noticing the needs of
others and responding.
What happens to the area and volume of 2D and 3D shapes when you
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?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
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?
What fractions of the largest circle are the two shaded regions?
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!
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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
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. . . .
Derive a formula for finding the area of any kite.
Determine the total shaded area of the 'kissing triangles'.
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. . . .
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).
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. . . .
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Can you maximise the area available to a grazing goat?
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?
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.
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
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.
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?
What do these two triangles have in common? How are they related?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
A simple visual exploration into halving and doubling.
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 rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
How would you move the bands on the pegboard to alter these shapes?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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. . . .
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.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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.