Plum Blossom
This problem was posted in Kazo City, Saitama Prefecture in 1881. Here is its entry at the Sangaku Archive [coming soon].
Five equal circles surround a central circle, as shown. If the central circle has diameter 17.55, what is the diameter of the surrounding circles?
Let the surrounding circles' radius be r, with y denoting the central circle's radius. The small right triangle, pictured above, has base and hypotenuse lengths r and r+y, respectively. To someone familiar with basic trigonometry, it is helpful to observe that this triangle's smallest angle is one-tenth of a full circle, or . Thus,
After some rearrangement,
Plugging in and , we see that . Therefore, the diameter of the surrounding circles is
The answer given on the tablet is 25. (The discrepancy might be due to coarse rounding or an error, neither of which are uncommon on sangaku.)
It is unlikely that 19th century Japanese mathematicians would have used trigonometry to solve this problem. First introduced to Japan by the Dutch in the mid-17th century, trigonometric tables found practical application to surveying, astronomy, and navigation. However, such tables (which included rounded decimal values of sine and cosine for integer-degree angles) were rarely used to solve sangaku problems. Scholars have several theories about why. One is that Japanese mathematicians lacked a well-developed notion of angle measure, preferring, instead, to reason in terms of arc length. Also, users of the tables were instructed to apply linear interpolation to estimate values between consecutive lines (e.g., ), thereby introducing additional imprecision. Evidently, these and other factors created a distaste among wasanka for trig tables.
On the other hand, this particular sangaku was dedicated 14 years into the Meiji period; it is not impossible that trigonometry might even have been used for abstract geometry problems around this time. Furthermore, Edo period mathematicians had discovered the equivalent of power series expansions for trigonometric functions, which in theory could be used to find an arbitrarily precise value of . (See Kitamuki Kannon for more on power series.) However, the technique used for this problem, as written on the tablet, suggests that neither a trig table nor a power series approximation was applied in this case. Instead, a trigonometry-free approach seems more likely.
Using similar triangles and some algebra, one can work out the ratio of a regular pentagon's diagonal (the segment connecting two non-adjacent vertices) to one of its sides:
(This quantity, known today as the golden ratio, makes occasional appearances in sangaku problems---even ones that don't involve pentagons.)
[[Link to Tashiro shrine sangaku]]
This gives us a right triangle with base r and hypotenuse equal to the golden ratio times . Using the Pythagorean theorem, we can determine that the length of segment AB is
Now applying Pythagoras to the smaller right triangle from before gives
which, when solved, yields
Plugging in gives an exact version of the earlier result. Using variations of techniques imported from China, one could compute these radicals as precisely (i.e., to as many digits) as desired.