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?
Explore one of these five pictures.
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?
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!
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
A follow-up activity to Tiles in the Garden.
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.
An investigation that gives you the opportunity to make and justify
A task which depends on members of the group noticing the needs of
others and responding.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Look at the mathematics that is all around us - this circular
window is a wonderful example.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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?
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
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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.
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?
What happens to the area and volume of 2D and 3D shapes when you
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
Can you work out the area of the inner square and give an
explanation of how you did it?
Determine the total shaded area of the 'kissing triangles'.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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. . . .
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?
How many tiles do we need to tile these patios?
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
What fractions of the largest circle are the two shaded regions?
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 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).
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. . . .
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. . . .
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?
Can you maximise the area available to a grazing goat?
These practical challenges are all about making a 'tray' and covering it with paper.
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.
This article for teachers gives some food for thought when teaching
ideas about area.
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?
A simple visual exploration into halving and doubling.
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Grandpa was measuring a rug using yards, feet and inches. Can you
help William to work out its area?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.