Sangaku in a Square

One of the simplest sangaku - geometric problems carved on colorful wooden tablets and offered in shinto shrines and buddhist temples during the Edo period (1603-1867) of self-imposed seclusion of Japan from the Western world - is presented by the following diagram:

A triangle is formed by a line that joins the base of a square with the midpoint of the opposite side and a diagonal. Find the radius of the inscribed circle.


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Copyright © 1996-2018 Alexander Bogomolny

A triangle is formed by two lines that join the base of a square with the midpoint of the opposite side and a diagonal. Find the radius of the inscribed circle.

A solution could be observed on an extended diagram. Let A, B, C, D be the vertices of the square and M the midpoint of AD. BD and CM intersect in P. Let Q be the intersection of BM and CD and H the projection of P on BC. The task is to find the inradius of ΔBCP.

Focus on ΔBCQ. MD is parallel to the base and is half as long which implies that this is a midline of the triangle. In other words, M and D are the midpoints of BQ and CQ, respectively. This means that BD and CM are two medians in the triangle and P its centroid. The centroid divides the medians in ratio 2:1 so that


CP = CM·2/3 and BP = BD·2/3.

Thus assuming BC = a, we can apply the Pythagorean theorem to first find BD (from ΔBCD) and CM (from ΔCDM) and subsequently, from (1) the sides of ΔBCP. By the same token, altitude PH = a·2/3. But in any triangle,


r·p = 2S,

where r is the inradius, p the perimeter, and S the area of the triangle. Putting everything together we see that

r·a·(1 + 5/2·2/3 + 2·2/3) = 2·a2/3,

from which r is easily found:


r = 2·a / (3 + 5 + 22).

The problem perhaps warrants a


The problem is clearly computational: the question is to find the inradius of a triangle related to a given square. Students may be tempted to use calculators or even dynamic geometry software to avoid handling square roots and fractions. This is what they might get - more or less. Assume a = 1. Then

BD = 1.414213562,
BP = .9428090416,
CM = 1.118033989,
CP = .7453559925,
p = 2.688165034,
QH = .6666666667,
S = .3333333333,
2S = .6666666666,
r = .2480006466,

which is quite in agreement with the exact answer (3). The difference between the two is that the numeric value, however accurate, carries no information as to the manner in which it was obtained. In (3), the appearance of the square roots is suggestive and points to a possible application of the Pythagorean theorem to two right triangles, as we did above. Thus (3) exhibits a valuable pattern that may trigger a thought process that may lead to the recollection of properties of medians and centroids, whereas r = .2480006466 in itself is a naked approximate number that, for all we know, is at best close enough to the exact answer.

In short, this 300 year old problem serves an example where the use of calculators actually obscures relationships between numbers. More such examples can be found in an edifying book What The Numbers Say by D. Niederman and D. Boyum along with a multitude of observations on everyday number usage and the ability to see beyond the numbers.


  1. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989

    Write to:

    Charles Babbage Research Center
    P.O. Box 272, St. Norbert Postal Station
    Winnipeg, MB
    Canada R3V 1L6

  2. D. Niederman and D. Boyum, What The Numbers Say, Broadway Books, 2003


  1. Sangaku: Reflections on the Phenomenon
  2. Critique of My View and a Response
  3. 1 + 27 = 12 + 16 Sangaku
  4. 3-4-5 Triangle by a Kid
  5. 7 = 2 + 5 Sangaku
  6. A 49th Degree Challenge
  7. A Geometric Mean Sangaku
  8. A Hard but Important Sangaku
  9. A Restored Sangaku Problem
  10. A Sangaku: Two Unrelated Circles
  11. A Sangaku by a Teen
  12. A Sangaku Follow-Up on an Archimedes' Lemma
  13. A Sangaku with an Egyptian Attachment
  14. A Sangaku with Many Circles and Some
  15. A Sushi Morsel
  16. An Old Japanese Theorem
  17. Archimedes Twins in the Edo Period
  18. Arithmetic Mean Sangaku
  19. Bottema Shatters Japan's Seclusion
  20. Chain of Circles on a Chord
  21. Circles and Semicircles in Rectangle
  22. Circles in a Circular Segment
  23. Circles Lined on the Legs of a Right Triangle
  24. Equal Incircles Theorem
  25. Equilateral Triangle, Straight Line and Tangent Circles
  26. Equilateral Triangles and Incircles in a Square
  27. Five Incircles in a Square
  28. Four Hinged Squares
  29. Four Incircles in Equilateral Triangle
  30. Gion Shrine Problem
  31. Harmonic Mean Sangaku
  32. Heron's Problem
  33. In the Wasan Spirit
  34. Incenters in Cyclic Quadrilateral
  35. Japanese Art and Mathematics
  36. Malfatti's Problem
  37. Maximal Properties of the Pythagorean Relation
  38. Neuberg Sangaku
  39. Out of Pentagon Sangaku
  40. Peacock Tail Sangaku
  41. Pentagon Proportions Sangaku
  42. Proportions in Square
  43. Pythagoras and Vecten Break Japan's Isolation
  44. Radius of a Circle by Paper Folding
  45. Review of Sacred Mathematics
  46. Sangaku à la V. Thebault
  47. Sangaku and The Egyptian Triangle
  48. Sangaku in a Square
  49. Sangaku Iterations, Is it Wasan?
  50. Sangaku with 8 Circles
  51. Sangaku with Angle between a Tangent and a Chord
  52. Sangaku with Quadratic Optimization
  53. Sangaku with Three Mixtilinear Circles
  54. Sangaku with Versines
  55. Sangakus with a Mixtilinear Circle
  56. Sequences of Touching Circles
  57. Square and Circle in a Gothic Cupola
  58. Steiner's Sangaku
  59. Tangent Circles and an Isosceles Triangle
  60. The Squinting Eyes Theorem
  61. Three Incircles In a Right Triangle
  62. Three Squares and Two Ellipses
  63. Three Tangent Circles Sangaku
  64. Triangles, Squares and Areas from Temple Geometry
  65. Two Arbelos, Two Chains
  66. Two Circles in an Angle
  67. Two Sangaku with Equal Incircles
  68. Another Sangaku in Square
  69. Sangaku via Peru
  70. FJG Capitan's Sangaku

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Copyright © 1996-2018 Alexander Bogomolny