1. 算 額 算 額

2. San Gaku are Japanese geometrical problems or theorems, originally found on wooden tablets which were placed as offerings at temples during the Edo period (1603–1867) by members of all social classes. 算額
San Gaku in Japanese:
It translates as ‘calculation tablet’

3. A selection of San Gaku problems
Find the relationship between the radii of the circles
The triangles are equilateral. Find the relationship between the radii of the circles
Find the relationship between the side of the square and the radii of the congruent circles
Find the relationship between the hypotenuse of the congruent right-angled triangles and the side of the regular pentagon.
Find the relationship between the sides of the coloured squares

4. Some of the problems required little more than Pythagoras’ theorem
The larger circles have a radius of 10cm.
What is the radius of the small circle?

5. The problems usually ask you to generalise
What is the relationship between the radius of the larger circles and the small circle?
radius b
radius b
radius a
Now try the four problems provided, firstly a numerical version, then an algebraic version

6. 1a) The congruent semi-circles have a radius of 5cm.
Find the length of the side of the square.

7. 1b) Find the relationship between the side of the square x and the radii r of the congruent semi-circles

8. 2a) The outline is a quarter-circle with radius 6cm
2a) The outline is a quarter-circle with radius 6cm. Joining the radii forms a rectangle as shown. Find the values of x and y
radius y
radius x

9. 2b) Prove that joining the radii forms a rectangle and find the relationship between the radii a, b and c
tangents perpendicular to radii
straight lines between radii and 4 right-angles ie rectangle
radius c
radius a
radius b

10. 3a) The square has 12cm sides. Find the radius of the circle
equate:

11. 3b) Find the relationship between the side of the square x and the radius of the circle r
equate:

12. 4a) Find the radius of the third circle
root and equate
radius 9
radius r

13. 4b) Find the relationship between the three radii
radius a
root and equate
divide by
radius b
radius c

14. Could you design a San Gaku of your own?
think those were tough?
Could you design a San Gaku of your own?