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WOLFRAM NOTEBOOK
Geometric Problems of Antiquity
Geometric Problems of Antiquity
Three geometric problems of antiquity were to square the circle, duplicate the cube, and trisect an angle, all using only a ruler and compass.
Squaring the circle means constructing a square with the same area as the circle. If the circle has radius 1, its area is and the square would have side length .
π
π
Duplicating the cube means constructing a cube with twice the volume of another cube. If the original cube has side length 1, its volume is 1, and the duplicating cube would have side length .
3
2
Trisecting an angle with angle measure means constructing a new angle with angle measure . Although some angles can be trisected, like 90, the problem is to be able to trisect any angle, not just special cases.
B
β
α=β/3
°
These constructions are possible using simple instruments other than a ruler and compass or given certain plane curves, and the numbers and can be approximated to any degree of accuracy. However, the constructions were all proved to be impossible with a ruler and compass, which does not stop some people from believing they have succeeded!
π
3
2