**Question**

**Plan of Attack**

This problem looks difficult because the shaded area does not seem to
look like any regular shape. Is it a
fish whose head is pointing in the top left direction and whose tail is in the
bottom left direction? Fortunately, this
is not a Rorschach ink-blot test.

As with all “area” problems in primary (elementary) school, we try to
break down the unfamiliar shape into regular shapes (e.g. parts of circles,
squares, triangles, rectangles). It is
basically a divide-and-conquer strategy (using heuristics H10 & H11). If we look carefully, we realise that the required
area consists of a semi-circle less a funny horn-shaped area, which I call ‘

*F*’.*F*is for*funny*, for want of a better description. So I am going to find the area of the semicircle (which is half of a circle), then subtract the area of*F*. We’ll worry about finding the area of*F*later.**Solution**

Area of
semi-circle [in cm

^{2}]
= ½ ´

*p*´ radius^{2}
= ½ ´

*p*´ (5)^{ 2}=^{25}/_{2}*p*
It is good
to leave the calculation with

*p*until the last step.
OK, we are done with the first part.
[Heuristic H11] Let us us tackle
the next part, which is to find the area of

*F*. Note that this is a 45°-45°-90° isosceles triangle minus a 45°-degree sector (which is one-eighth of a circle, because^{45°}/_{360°}=^{1}/_{8}).
Area of

*F*[in cm^{2}]
= Area of triangle – area of sector

= ½ ´ 10 ´ 10 –

^{1}/_{8}´*p*´ radius^{2}
= ½ ´ 10 ´ 10 –

^{1}/_{8}´*p*´ (10)^{ 2}
= 50 –

^{25}/_{2}*p*Let us combine our answers. We need to subtract 50 –

^{25}/

_{2}

*p*from the area of the semi-circle. If we subtracted 50 from

^{25}/

_{2}

*p*, we would have over-subtracted. So we need to add back

^{25}/

_{2}

*p*. Hence

Required
Area [in cm

^{2}]
= Area of semi-circle – area of

*F*
=

^{25}/_{2}*p*– (50 –^{25}/_{2}*p*)
=

^{25}/_{2}*p*– 50 +^{25}/_{2}*p*
= 25

*p*– 50
Using the calculator’s value of

*p*, we obtain
Required
Area = 28.54 cm

^{2}(to 2 decimal places)
H10. Simplify the problem

H11. Solve part of the problem

**Commentary**

This
difficult problem was solved by dividing the problem into smaller pieces and
tackling each piece one at a time. We
break down a complicated shape into familiar shapes. That is the secret.

Please refer to this similar problem.

Please refer to this similar problem.

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