Square and Circle in a Gothic Cupola

This is another in a series of simple sangaku problems that is solved by a repeated application of the Pythagorean theorem.

Two quarter circles inscribed in a square form a gothic cupola. Inscribed in the latter is a circle on top which stands a small circle. What is the relation between the radius of the circle and the side of the small square and the side the big square?

Solution

This is another in a series of simple sangaku problems that is solved by a repeated application of the Pythagorean theorem.

Solution

With the reference to the diagram, assume the side of the big square is 1, x the side of the small square and r is the radius of the circle in question.

In right ΔAFG, AG² = AF² + FG²:

1 = (1/2 + x/2)₂ + x₂,

which simplifies to a quadratic equation

5x² + 2x - 3 = 0,

with a single positive root, x = 3/5. The Pythagorean theorem applied to ΔAMO, AO² = AM² + MO², supplies an equation for the radius r:

(1 - r)² = (1/2)² + (x + r)²,

which, with x = 3/5, simplifies to a linear equation in r so that r = 39/320.

References

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

Sangaku

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