Square root of 6
The square root of 6 is the positive real number that, when multiplied by itself, gives the natural number 6. It is more precisely called the principal square root of 6, to distinguish it from the negative number with the same property. This number appears in numerous geometric and number-theoretic contexts.
It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:
which can be rounded up to 2.45 to within about 99.98% accuracy.
Since 6 is the product of 2 and 3, the square root of 6 is the geometric mean of 2 and 3, and is the product of the square root of 2 and the square root of 3, both of which are irrational algebraic numbers.
NASA has published more than a million decimal digits of the square root of six.
File:Slide rule with square roots of 6 and 7.jpg|thumb|A Logarex system Darmstadt slide rule with 7 and 6 on A and B scales, and square roots of 6 and of 7 on C and D scales, which can be read as slightly less than 2.45 and somewhat more than 2.64, respectively
Geometry
In plane geometry, the square root of 6 can be constructed via a sequence of dynamic rectangles, as illustrated here.In solid geometry, the square root of 6 appears as the longest distances between corners of the double cube, as illustrated above. The square roots of all lower natural numbers appear as the distances between other vertex pairs in the double cube.
The edge length of a cube with total surface area of 1 is or the reciprocal square root of 6. The edge lengths of a regular tetrahedron, a regular octahedron, and a cube of equal total surface areas satisfy.
The edge length of a regular octahedron is the square root of 6 times the radius of an inscribed sphere.