Squaring the circle
Problem
Bluey - green squares, white squares, 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 an estimate. Note the totals before embarking on a more rigorous audit of what is there.
How accurate was your estimate?
Can you give an upper and a lower bound to your estimate?
If the blue-green squares are of 1 and 1.5 units of length, the white squares of 1.5 units and the transparent squares 1.75 units of length - how "big" is the circular part of the pavement?
Getting Started
Plenty to keep you busy here.
You can see a full quarter of the circle and you can extrapolate from this fairly accurately.
Student Solutions
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:
| Type of piece and location | Number of pieces in a quarter circle |
|---|---|
| Big transparent square, inner part | 47 |
| White rectangular pieces, inner part | 94.5 |
| Blue-grey squares, inner part | 54 |
| White squares, circumference | 28 |
| Blue-grey squares, circumference | 29 |
To determine the total number from each type of pieces, I have to multiply all results by 4:
| Type of piece and location | Number of pieces in a whole circle |
|---|---|
| Big transparent square, inner part | 188 |
| White rectangular pieces, inner part | 378 |
| Blue-grey squares, inner part | 216 |
| White squares, circumference | 112 |
| Blue-grey squares, circumference | 116 |
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 estimate.
The pavement is a circle, of radius
20 units (inner part) + 2.5 units (circumference) = 22.5 units
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:
| Type of piece and location | Unit area (sq units) | Total area (sq units) |
|---|---|---|
| Big transparent square, inner part | 2.25 | 423 |
| White rectangular pieces, inner part | 1.5 | 567 |
| Blue-grey squares, inner part | 1 | 216 |
| White squares, circumference | 1.5625 | 175 |
| Blue-grey squares, circumference | 1.5625 | 181.25 |
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 before.