Henry Ernest Dudeney/Puzzles and Curious Problems/188 - Squaring the Circle/Solution 1
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Puzzles and Curious Problems by Henry Ernest Dudeney: $188$
- Squaring the Circle
- The problem of squaring the circle depends on finding the ratio of the diameter to the circumference.
- This cannot be found in numbers with exactitude,
- but we can get near enough for all practical purposes.
- But it is equally impossible, by Euclidean geometry, to draw a straight line equal to the circumference of a given circle.
- You can roll a penny carefully on its edge along a straight line on a sheet of paper and get a pretty exact result,
- but such a thing as a circular garden-bed cannot be so rolled.
- Now, the line below, when straightened out
- (it is bent for convenience in presentation),
- is very nearly the exact length of the circumference of the accompanying circle.
- The horizontal part of the line is half the circumference.
- Could you have found it by a simple method, using only pencil, compasses and ruler?
Solution
Make a rectangle with one side equal to the diameter and one side $3$ times the diameter.
Then take the diagonal of this rectangle.
This will be quite close to the circumference, as it is $\sqrt {10}$ or approximately $3.162$ times the diameter.
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $188$. -- Squaring the Circle
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $289$. Squaring the Circle