How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
Can you work out the area of the inner square and give an
explanation of how you did it?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
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 find the area of a parallelogram defined by two vectors?
An investigation that gives you the opportunity to make and justify
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?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
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.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
A follow-up activity to Tiles in the Garden.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Explore one of these five pictures.
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?
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?
What shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
How many tiles do we need to tile these patios?
A simple visual exploration into halving and doubling.
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?
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.
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?
Can you maximise the area available to a grazing goat?
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.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
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?
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.
This article, written for teachers, discusses the merits of
different kinds of resources: those which involve exploration and
those which centre on calculation.
What do these two triangles have in common? How are they related?
I cut this square into two different shapes. What can you say about
the relationship between them?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
This article for teachers gives some food for thought when teaching
ideas about area.
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
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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.
Derive a formula for finding the area of any kite.
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?
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