Sangaku Math May 2026

1. What Are Sangaku? Sangaku (算額, literally "calculation tablet") are colorful wooden tablets depicting geometric problems, often solved and dedicated to Shinto shrines or Buddhist temples in Japan. They were created by people from all walks of life—samurai, farmers, merchants, and professional mathematicians (called wasanka )—from the early 17th to the late 19th century (the Edo period).

Place the line as the x-axis: (y=0). Let circle (R) have center ((R, R)) — it touches the line at ((0,0)). Let circle (r) have center ((d, r)) with (d > 0), touching the line at ((d, 0)). sangaku math

From first equation: [ (h - R)^2 + (x - R)^2 = (R + x)^2 ] [ (h - R)^2 + x^2 - 2Rx + R^2 = R^2 + 2Rx + x^2 ] [ (h - R)^2 - 2Rx = 2Rx ] [ (h - R)^2 = 4Rx ] [ h - R = 2\sqrt{Rx} \quad \Rightarrow \quad h = R + 2\sqrt{Rx} ] They were created by people from all walks

Center = ((h, x)), tangent to line at ((h,0)). Tangency to circle (R): distance between centers = (R + x): [ \sqrt{(h - R)^2 + (x - R)^2} = R + x ] Tangency to circle (r): distance between centers = (r + x): [ \sqrt{(h - (R+2\sqrt{Rr}))^2 + (x - r)^2} = r + x ] Let circle (r) have center ((d, r)) with

Distance between centers of (R) and (r) = (R + r) (external tangency): [ \sqrt{(d-R)^2 + (r-R)^2} = R + r ] Simplify: [ (d-R)^2 + (r-R)^2 = (R+r)^2 ] [ (d-R)^2 + R^2 - 2Rr + r^2 = R^2 + 2Rr + r^2 ] [ (d-R)^2 - 2Rr = 2Rr ] [ (d-R)^2 = 4Rr ] [ d - R = 2\sqrt{Rr} \quad (\text{positive since } d > R) ] [ d = R + 2\sqrt{Rr} ]