Square Root - Geometric Construction of The Square Root

Geometric Construction of The Square Root

A square root can be constructed with a compass and straightedge. In his Elements, Euclid (fl. 300 BC) gave the construction of the geometric mean of two quantities in two different places: Proposition II.14 and Proposition VI.13. Since the geometric mean of a and b is, one can construct simply by taking b = 1.

The construction is also given by Descartes in his La Géométrie, see figure 2 on page 2. However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.

Euclid's second proof in Book VI depends on the theory of similar triangles. Let AHB be a line segment of length a + b with AH = a and HB = b. Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH as h. Then, using Thales' theorem and as in the proof of Pythagoras' theorem by similar triangles, triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don't need that but it is the essence of the proof of Pythagoras' theorem) so that AH:CH is as HC:HB i.e. from which we conclude by cross-multiplication that and finally that . Note further that if you were to mark the midpoint O of the line segment AB and draw the radius OC of length then clearly OC > CH i.e. (with equality when and only when a = b), which is the arithmetic–geometric mean inequality for two variables and, as noted above, is the basis of the Ancient Greek understanding of "Heron's method".

Another method of geometric construction uses right triangles and induction: can, of course, be constructed, and once has been constructed, the right triangle with 1 and for its legs has a hypotenuse of . The Spiral of Theodorus is constructed using successive square roots in this manner.