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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Monday, May 4, 2015
[Pri20150503SMT] Seeing through the Area of Mess
like a challenging problem regarding area.
The diagram looks very confusing.
The shaded area consists of many convex pieces. Furthermore, the unshaded regions comprise
triangle-like pieces of which we do not know all the dimensions. The only dimensions we know is related that
of the large triangle DACB. What shall we do?
The key to
solving this problem is to appropriately cut up the figure so as to be able to
“see” it properly. We can cut up the
figure like this:-
might note that the pink triangles and green triangles are right-angled
triangles, because they are in semi-circles.
Yeah! Smart! But how can this be useful? We do not know their individual bases and
heights. We only know their longest
sides. Hmmmmm ... Ah! However,
the pink triangles and
green triangles all add up to the large triangle DACB. This is a key observation.
Note that the convex parts can be viewed
as semi-circles with either a pink triangle or green triangle taken away. There are two pairs of semi-circles: one
large and one small. Therefore, the
shaded area is the total of one small circle plus one large circle minus the total
of the areas of the pink and green triangles (which is the same as the area of
triangle DACB). As you know,
the area of the triangle DACB is ½ ´ base ´ height, in which ½ ´ base = 5 cm. Once
you understand all these, the calculation is very easy.
total area of
= area of large circle + area of small circle – area of