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
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
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.
These constructions are possible using simple instruments other than a ruler and compass or given certain plane curves, and the numbers
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!