A blog about Mathematics and Mathematics Education in Singapore, as well as Mathematics Education in general. Written for students, parents, educators and other stakeholders in Singapore, and around the world.
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Saturday, May 2, 2015
[Pri20150408YYS] Yin-Yang Semicircles?
looks a bit like the Yin and Yang symbol, doesn’t it? This problem can be solved easily using the
correct insights. I present two solutions:
the first one is by direct calculation (in terms of p), and the second solution uses the powerful concept
of ratio of similar figures.
method is used, first we must make observations. Can you see that there are three types of
semicircles (small, medium and large)? [H02, H04]
Let S = area of small semicircle, M =
area of medium-sized semicircle,
and L = area of large semicircle.
Note that area of A
= area of C = L
– M + S, and area of B
= 2´(M – S). [H10, H11]
Solution 1 (by direct calculation)
= ½p(3)2 = 9p/2. M
= ½p(2)2 = 4p/2. S
= ½p(1)2 = p/2.
area of A
= area of C = 9p/2
– 4p/2 + p/2 = 3p
area of B = 2´(4p/2
– p/2) = 3p
\ area of A :
area of B : area of C = 1 : 1 : 1. (The areas are all the same)
Many pupils feel more
comfortable using concrete approximations like
p»22/7 or p» 3.14, but this tends to obscure relationships
between entities, and makes the calculations messier.
Solution 2 (using similar shapes)
powerful idea is that the ratio of areas of similar shapes is the square of the
ratios of their lengths. When a figure
is enlarged by a factor of (say) 5, we get a similar figure and the area
becomes 52 = 25 times as
large. So M = (2)2S = 4S
because the radius of the medium-sized semicircle is twice that of the
small semicircle. Likewise, L = (3)2S = 9S. area of A
= area of C = 9S
– 4S + S = 6S
area of B = 2´(4S – S) =
\ area of A :
area of B : area of C
= 1 : 1 : 1.
(The areas are all the same)
solution is neater because we do not need to deal with fractions or with p. The ratio of areas of similar shapes is the square
of the ratios of their lengths. This concept is in fact required knowledge in secondary
school mathematics, including GCE ‘O’ level “Elementary” Mathematics.