Japan “Sangaku” Research Institute

Category: Sangaku

Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (04)

Problem As shown in the figure, if there is a round field with a diameter of $100 \ m$ and it is divided equally among $5$ people, find the length of each of the red and blue line segments. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$…

March 17, 2024
Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (03)

Problem As shown in the figure below, two equilateral triangles $ACE$ and $BDF$ are inscribed in a regular hexagon $ABCDEF$, and a great circle with a diameter of $1 \ m$ is inscribed in them. If a small circle with a diameter of $t \ m$ is inscribed in an isosceles triangle $ABF$, and small…

March 10, 2024
Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (02)

Problem There is a right triangular piece of land with a short side of $30\ m$ and a long side of $40 \ m$. As shown in the diagram, add $2 \ m$ wide roads to this, and divide the remaining area equally into three sections, $S_1, S_2$, and $S_3$. What are their shapes? And…

March 3, 2024
Katayama-hiko Shrine (1873), Osafune-cho, Setouchi City, Okayama Prefecture (01)

Katayama-hiko Shrine is located $850$ meters southeast from Osafune Station on the JR Ako Line. Problem As shown in the figure, two mutually circumscribed circles of diameter $d$ are inscribed in a fan of radius $R$. $c$ is the common tangent of these two circles, and is also the chord of the fan that inscribes…

February 27, 2024
Koh-jingu Shrine (1880), Higashiaso, Soja City, Okayama Prefecture (04)

Problem If there is a cone, its generating line is $a$, and the slope angle is $α$, find the area of the side surface. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution The…

February 5, 2024
Koh-jingu Shrine (1880), Higashiaso, Soja City, Okayama Prefecture (03)

Problem If a person’s moon shadow is reflected in the water, its length is $10 ft$ from the feet, and the height from the water’s surface to the eyes is $6 ft$, How many degrees does the moon hang over? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…

January 29, 2024
Koh-jingu Shrine (1880), Higashiaso, Soja City, Okayama Prefecture (02)

Problem If the distance of $1°$ at the equator is $28.03 \ ri$, how many degrees north latitude is the place where the distance of $1°$ is $27 \ ri$ ? ($1 \ ri$ is approximately $3.927 \ km$.) $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$…

January 22, 2024
Koh-jingu Shrine (1880), Higashiaso, Soja City, Okayama Prefecture (01)

Koh-jingu Shrine is located $1.8 \ km$ north of Ashimori Station on the JR Kibi Line. Problem If the distance of $1°$ at the equator is $28.03 \ ri$, how much is the distance of $1°$ at $35°$ north latitude? ($1 \ ri$ is approximately $3.927 \ km$.) $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$…

January 16, 2024
Hachiman Wake Shrine (1890), Sako, Akaiwa City, Okayama Prefecture (24)

Problem What is the result when you add $\dfrac{3}{5}$ to $\dfrac{1}{9}$? $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution $$\frac{1}{9}+\frac{3}{5}=\frac{5}{45}+\frac{27}{45}=\frac{32}{45}.$$ $ $ $ $ $ $ $ $ $$Ans. \quad \frac{32}{45}$$ Reference Yoshikazu…

December 31, 2023
Hachiman Wake Shrine (1890), Sako, Akaiwa City, Okayama Prefecture (23)

Problem What is the sixth root of $59604644775390625$ $\ ?$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution When $59604644775390625$ is factorized into prime factors, $$59604644775390625=5^{24}=(5^4)^6=625^6.$$ Therefore, we find that $$\sqrt[6]{59604644775390625}=625.$$ $ $…

December 31, 2023