Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
What fractions of the largest circle are the two shaded regions?
An investigation that gives you the opportunity to make and justify
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?
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 help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
How many centimetres of rope will I need to make another mat just
like the one I have here?
What happens to the area and volume of 2D and 3D shapes when you
These practical challenges are all about making a 'tray' and covering it with paper.
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Explore one of these five pictures.
Can you draw a square in which the perimeter is numerically equal
to the 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
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
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
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 thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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 maximise the area available to a grazing goat?
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
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
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?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
A follow-up activity to Tiles in the Garden.
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 work out the area of the inner square and give an
explanation of how you did it?
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
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .