算額算額.

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Presentation transcript:

算額算額

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’

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

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?

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

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

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

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

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

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

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

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

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

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