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 fractions of the largest circle are the two shaded regions?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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?
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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
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 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
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 is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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?
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.
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!
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?
What happens to the area and volume of 2D and 3D shapes when you
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
A task which depends on members of the group noticing the needs of
others and responding.
Explore this interactivity and see if you can work out what it
does. Could you use it to estimate the area of a shape?
Can you maximise the area available to a grazing goat?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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?
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 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. . . .
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. . . .
Determine the total shaded area of the 'kissing triangles'.
Can you work out the area of the inner square and give an
explanation of how you did it?
Derive a formula for finding the area of any kite.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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).
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
An investigation that gives you the opportunity to make and justify
Look at the mathematics that is all around us - this circular
window is a wonderful example.
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 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
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?
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
Use the information on these cards to draw the shape that is being described.