Here's something related to a primary school math problem in Singapore that has gotten me a little stumped. I'm not too bothered by this since I'm trying to have fun reasoning about it from first principles. Given the diagram below, where the arcs represent quadrants of a circle of radius
r:
Is there a way to figure out the areas of A, B and C in terms of r in such a way that if r is provided, we are able to solve for A, B and C?
Here's the original problem as context. I could not solve for
A (the same in both contexts) somehow. At the very least, I felt like I needed to be able to solve for
C or
D (
D &
B having the same areas because of symmetry, so
B is a red herring). In the above case, I may have made it more difficult by breaking
C and
D (in the original) into their symmetric component parts. As an orthogonal question, I wonder about the proof of symmetry.
1. Find the area of B+C (trivial: Quarter the area of the circle)
2. Find the area of A (stumped)
3. Find the area of C-A (Needs some work, but not especially hard to find from first principles. That was what led to my drawing which completed all the symmetries)
4. Find the area of A+B (trivial: Area of square minus the quadrant)
Solution (via Trigonometry) - updated 7/4
Okay, so I cheated. I had intended to see if trigonometry could generate another equation that would allow
A to be determined based solely on
r and what we know about the areas of squares and circles. Well, trigonometry provides a direct answer:
There are two ways of determining the angles, the more direct being the radius of each arc. The angles are
60 and
30 degrees, with the height of the central triangle being
sqrt(3) x 7. So, the area of the central triangle is
sqrt(3) x 49. Each pie slice has area
(1/12) x pi x 196. So the area of
A is
196 - ((sqrt(3) x 49) + (1/6 x pi x 196)). Ignoring the original use of
22/7 as
pi (which results in an error of
0.625 when computing the area of the two pie slices), we get the area of 8.50415041 for
A. I do not believe there is a convenient integer representation of
sqrt(3) in terms of
pi, so I think I am prepared to call "bollocks" on the teacher(s) who thought
A could be conveniently found in the original problem! Of course, I am also prepared to be wrong.
I am currently working on a sanity check. We know
C - A =
112 (with
pi =
22/7). So, I am trying to independently derive this answer through trigonometry.
Okay, sanity check done.
C can be determined independently of
A. There is a relationship between
C, the 60-degree pie slice and the equilateral triangle in
C. That works out to
120.38023.
C - A in the original problem can be calculated to be
111.87608 by avoiding the lousy estimate of
22/7 for pi. That was
196 - (2 x (1/4 x pi x 196)).
C - A computed using the direct determination of
A and
C via trigonometry yields exactly the same answer.
So I'm definitely calling bollocks on those teachers.
Final Comment: Of course, through trigonometry, I have answered my first question - yes, we can determine A, B and C in terms of r.
