# Pentagon Proportions Sangaku

Due to the design of my logo, this simple sangaku is quite dear to my heart. (The problem was described and solved in volume 3 of a 1810 text by Tenshoho Sinan and Anmei Aida, *Geometry and Algebra*, see [Fukagawa & Pedoe, Example 4.3].)

Construct a regular pentagon by tying a knot in a strip of paper of width a. Calculate the side t of the pentagon in terms of a.

The method of construction is "right over left and under". The paper is then smoothed out, and the pentagon appears. *Billet doux* (a love letter or note) were sent thus in Japan.

### References

H. Fukagawa, D. Pedoe,

*Japanese Temple Geometry Problems*, The Charles Babbage Research Center, Winnipeg, 1989Write to:

Charles Babbage Research Center

P.O. Box 272, St. Norbert Postal Station

Winnipeg, MB

Canada R3V 1L6

## Sangaku

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- Critique of My View and a Response
- 1 + 27 = 12 + 16 Sangaku
- 3-4-5 Triangle by a Kid
- 7 = 2 + 5 Sangaku
- A 49
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- A Hard but Important Sangaku
- A Restored Sangaku Problem
- A Sangaku: Two Unrelated Circles
- A Sangaku by a Teen
- A Sangaku Follow-Up on an Archimedes' Lemma
- A Sangaku with an Egyptian Attachment
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- An Old Japanese Theorem
- Archimedes Twins in the Edo Period
- Arithmetic Mean Sangaku
- Bottema Shatters Japan's Seclusion
- Chain of Circles on a Chord
- Circles and Semicircles in Rectangle
- Circles in a Circular Segment
- Circles Lined on the Legs of a Right Triangle
- Equal Incircles Theorem
- Equilateral Triangle, Straight Line and Tangent Circles
- Equilateral Triangles and Incircles in a Square
- Five Incircles in a Square
- Four Hinged Squares
- Four Incircles in Equilateral Triangle
- Gion Shrine Problem
- Harmonic Mean Sangaku
- Heron's Problem
- In the Wasan Spirit
- Incenters in Cyclic Quadrilateral
- Japanese Art and Mathematics
- Malfatti's Problem
- Maximal Properties of the Pythagorean Relation
- Neuberg Sangaku
- Out of Pentagon Sangaku
- Peacock Tail Sangaku
- Pentagon Proportions Sangaku
- Proportions in Square
- Pythagoras and Vecten Break Japan's Isolation
- Radius of a Circle by Paper Folding
- Review of Sacred Mathematics
- Sangaku à la V. Thebault
- Sangaku and The Egyptian Triangle
- Sangaku in a Square
- Sangaku Iterations, Is it Wasan?
- Sangaku with 8 Circles
- Sangaku with Angle between a Tangent and a Chord
- Sangaku with Quadratic Optimization
- Sangaku with Three Mixtilinear Circles
- Sangaku with Versines
- Sangakus with a Mixtilinear Circle
- Sequences of Touching Circles
- Square and Circle in a Gothic Cupola
- Steiner's Sangaku
- Tangent Circles and an Isosceles Triangle
- The Squinting Eyes Theorem
- Three Incircles In a Right Triangle
- Three Squares and Two Ellipses
- Three Tangent Circles Sangaku
- Triangles, Squares and Areas from Temple Geometry
- Two Arbelos, Two Chains
- Two Circles in an Angle
- Two Sangaku with Equal Incircles
- Another Sangaku in Square
- Sangaku via Peru
- FJG Capitan's Sangaku

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Copyright © 1996-2018 Alexander Bogomolny

Construct a regular pentagon by tying a knot in a strip of paper of width a. Calculate the side t of the pentagon in terms of a.

### Solution

Label the points and lengths as in the diagram below:

Since ∠BCF = ∠CAM

BF : BC = CM : AC,

or

s : t = t/2 : p.

But p = t + 2s, from which

t = (1 + √5) s.

(Since ∠BCF = 18°, this just says that

From BC² = BF² + FC², we successively get

t² = [t / (1 + √5)]² + a²,

a² = [(5 + √5) / 8] · t²,

a = [√10 + 2√5 / 4] · t,

t = √2 - 2√(4/5) · a.

**Note**: after the above derivation Fukagawa and Pedoe discus an elegant construction of a regular pentagon with straight edge and compass.

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Copyright © 1996-2018 Alexander Bogomolny

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