What fractions of the largest circle are the two shaded regions?
An investigation that gives you the opportunity to make and justify
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?
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. . . .
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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
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 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.
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?
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?
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?
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?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
How many centimetres of rope will I need to make another mat just
like the one I have here?
Can you work out the area of the inner square and give an
explanation of how you did it?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Explore one of these five pictures.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
A follow-up activity to Tiles in the Garden.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
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. . . .
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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!
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?
How many tiles do we need to tile these patios?
A task which depends on members of the group noticing the needs of
others and responding.
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?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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
What happens to the area and volume of 2D and 3D shapes when you
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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. . . .
Choose a box and work out the smallest rectangle of paper needed to
wrap it so that it is completely covered.
Determine the total shaded area of the 'kissing triangles'.
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. . . .