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?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Can you work out the area of the inner square and give an
explanation of how you did it?
Here is a solution sent in by Andrei of School 205, Bucharest.
Does anyone think they can do better? Well done Andrei.
I first looked up to find the quarter square in the figure of
the problem. From reasons of symmetry, I delimitated it, and the
result is in the figure below, where the two perpendicular
diameters are in red:
This procedure was necessary because the figure does not give
the whole circle.
I counted as precise as I could each type of pieces, including
parts of pieces on the borders. The results, for the quarter of
circle, are in the table below:
To determine the total number from each type of pieces, I have
to multiply all results by 4:
I think that my estimate is accurate enough, because
I tried to count carefully quarter of pieces on the border and to
compensate big parts with small ones. My estimate must be more
accurate than 5%.
Now I calculate the area of the pavement, and the areas covered
by each colour. This way I shall see the accuracy of my previous
The pavement is a circle, of radius
20 units (inner part) + 2.5 units (circumference) = 22.5
The area of the pavement is the area of the circle, of radius
22.5 units, i.e. 1590 sq units..
Now I calculate the area covered by pieces of each colour:
Now, the total area of the pavement, obtained by adding the
numbers from the last column of the table, is 1562.25 sq units.
Comparing this with the result 1590 sq units obtained previously, I
find an accuracy of around 1.7%, even higher than what I estimated