Pi approximation without words
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Archimedes found that π≈22/7… But could you explain that approximation intuitively?
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Archimedes found that π≈22/7… But could you explain that approximation intuitively?
I provide a “proof without words” that is easy to print and easy to understand:
The pieces form a 22×7 cm rectangle exactly, but could also form an irregular dodecagon that approximates a 7 cm radius circle. So the area of a circle, which is known to be πR², is almost equal to (22/7)R² and, therefore, π≈22/7.
In the same image, two radii of the circle are marked. One of them is 7 cm long (exactly). The other one is 5√2 cm long (approximately). So 7≈5√2 and, therefore, √2≈7/5.
The size of the pieces can be deduced from the rectangle picture:
The right triangles legs measure 2 cm and 3.25 cm.
The rectangle sides measure 2 cm and 3.5 cm.
The irregular hexagons fit in a 10×7 cm rectangle.