Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
A man paved a square courtyard and then decided that it was too
small. He took up the tiles, bought 100 more and used them to pave
another square courtyard. How many tiles did he use altogether?
Can you work out the area of the inner square and give an
explanation of how you did it?
This problem was a rather special one, because many of you would
not have learnt about pi yet at school.
This solution by Rebeccca Jackson (11 years)
from the Sum Fun Maths Club (Stamford High School, Lincolnshire)
sets out the steps for solving the problem very clearly. Rebecca
has used 3.14 for pi, as was suggested in the problem, and has
Christina Ivanova (London) agreed with this
George Vassilev (10 years, Rosebank Primary,
Leeds) sent in a well written summary of how he worked out this
"The first thing I did was to find a side of the square = 4.4. I
found that by 2.2 x2cm. which is the radius time 2. After that I
found out the area of the square: 4.4cm x 4.4cm = 19.36cm
2. Afterwards I found the area of the circle: 4.14 x 2.2
x 2.2 = 15.1976cm 2. Then I subtracted them to find out
what area of the square is not covered by the circle: 19.36 -
15.1976 = 4.1624cm 2."
This excellent solution was sent in by Syed Farhan
Iskandar, (12 years, Foxford School and Community
College). It doesn't actually state what value was used for pi
though, but it must have had 10 decimal places. This is why this
answer is slightly different to some others.
Radius of circle x 2 = length and width of each sides of the
2.2 cm x 2 = 4.4 cm
Length x width = area of square
4.4 cm x 4.4 cm = 19.36 cm 2
Area of square is 19.36 cm
pr 2 = area of circle
p x 2.2 cm 2 = 15.20530844 cm 2
Area of circle is 15.20530844 cm
Area of square - area of circle = area of square-not covered by
19.36 cm 2 - 15.20530844 cm 2 = 4.15469156
Area of square-not covered by circle is 4.15469156 cm