[Pri20150503SMT] Seeing through the Area of Mess

Question

Introduction

     This looks 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:-

     You 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.

Solution

   total area of shaded regions

= area of large circle + area of small circle – area of triangle DACB

= [p (5)² + p (⁵/₂)² – 5 ´ 10] cm²

= 48.18 cm²

Heuristics Used

H02. Use a diagram / model

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

Suitable Levels
* Primary School Mathematics
* other syllabuses that involve area of triangles and circles