First, the solution (might be obvious to many, but more on that later). The shaded (blue) area can be considered as being made up of two areas:

1. The area outside the circles
2. the area within the (center) circle

For the shaded area outside the circles, we can observe by symmetry that it is exactly half of the total area outside the circles. As for the shaded area inside the center circle, it is half of the area of that circle, again by symmetry.

Therefore, we have:

Shaded Area = 0.5(Area outside circles)+0.5(Area of a circle)
= 0.5(Area of rectangle - 3*Area of a circle)
+0.5(Area of a circle)
= 0.5*Area of rectangle - Area of a circle
= 0.5*(36cm*12cm) - π*(6cm)²
= (216 - 36π)cm²
= 102.90cm²
[to two decimal places (what's the convention now huh?)]

Fairly trivial - of more interest might be what approaches won't work. What would students not seeing the symmetry between the pieces do? Perhaps we should go the whole hog, and simulate where various levels of competency would get them, with reference to a (slightly-dated) syllabus.

I. Basics
(Primary Three) Division (of 36cm) by three (for the diameter of the circles), and further by two (for the radius and areas)
(Primary Three) Calculate the area of a rectangle
(Primary Six) Calculate the area of a circle

Let's face it, we ain't getting very far if the student hasn't grasped the meaning of "divide" or "area".

II. Symmetry
(Primary Four) Identifying symmetric figures, etc

Note that the concept of symmetry appears to be covered only in Primary Four, and that rotational symmetry is not even included. Strictly speaking, though, being able to identify symmetric figures is enough - it's just a matter of knowing what to look for, in particular that the twelve pieces outside the circles can be seen as six pairs:



Observe how the six areas A, B, C, D, E and F are exactly the same, each with one shaded and one unshaded version (the areas Z could be considered as a distraction to the unwary); any piecewise attempt to get the area by independently calculating the areas of the six areas and adding them would almost certainly fail. Some like B and C are doable, by trying to get A or D by themselves would take far more advanced math than any schoolkid should be expected to master.


Question is: is this obvious? Perhaps not.



While perhaps unfathomable by those experienced in math (and the more demanding teachers/parents, regardless of their own facility in math), this "blindness" can be understood as being similar to not perceiving the faces in the left side of the above image. I daresay a fair number of those who have never seen the illusion before would not be able to pick out the "two faces" from the left half, and some maybe not even from the right half, where the vase detail is removed.

Does it mean that those unable to see the faces are stupid? Not quite; it is not difficult to imagine an otherwise clever fellow, who just doesn't "get" it, for some particular image (as with the Internet's "when you see it..." collections). While not everybody may be equipped to be a Newton or Gauss, very few come anywhere close to maximizing their potential (but yes, it's fair for the less-devoted to ask, what's the point?)

Back to the question. Actually, why does the shaded area A have the same area as the unshaded area A? There are several approaches, beginning with the intuition that "err... well they look the same when I rotate them in my mind, and they all fit, so..." (the spirit of this form of inquiry sometimes extended to having an epiphany that some particular god exists, as deliciously picked on in Wisconsin). It happens to be true in this case, but not always:


This illusion has been around for a long time, but only got popular when chocolate was used as the demonstration material. I wonder why.

As for myself, while I often have cause to be galled at how lousy at math I actually am (let's face it, 99% of work before higher-level university courses are just "how to calculate", and not the real thing), the shame has not been enough for me to seek solid proofs instead of going "the engineering way":


If it's any consolation, they end up in the same place.

Joseph Chinnock - [bert's blog]

property valuation - [bert's blog]

personal finance blogs for 20 somethings - [bert's blog]

personal finance company harriman tn - [bert's blog]

personal finance blogs 2014 - [bert's blog]

property valuer - [bert's blog - Adding The Sums]

รับทำ seo - [bert's blog]