Square root of 3


The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as 3 or 31/2. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as Theodorus' constant, after Theodorus of Cyrene, who proved its irrationality.

Square root of 3
Equilateral triangle with side 2.svg
The height of an equilateral triangle with sides of length 2 equals the square root of 3.
Continued fraction

As of December 2013, its numerical value in decimal notation had been computed to at least ten billion digits.[1] Its decimal expansion, written here to 65 decimal places, is given by OEISA002194:


The fraction 97/56 (1.732142857...) can be used as an approximation. Despite having a denominator of only 56, it differs from the correct value by less than 1/10,000 (approximately 9.2×10−5). The rounded value of 1.732 is correct to within 0.01% of the actual value.

The fraction 716035/413403 (1.73205080756...) is accurate to 10^-11.

Archimedes reported a range for its value: (1351/780)2
> 3 > (265/153)2
;[2] the lower limit accurate to 1/608400 (six decimal places) and the upper limit to 2/23409 (four decimal places).


It can be expressed as the continued fraction [1; 1, 2, 1, 2, 1, 2, 1, …] (sequence A040001 in the OEIS).

So it's true to say:


then when   :


It can also be expressed by generalized continued fractions such as


which is [1; 1, 2, 1, 2, 1, 2, 1, …] evaluated at every second term.

Geometry and trigonometryEdit

The height of an equilateral triangle with edge length 2 is 3. Also, the long leg of a 30-60-90 triangle with hypotenuse 2.
And, the height of a regular hexagon with sides of length 1.
The diagonal of the unit cube is 3.
This projection of the Bilinski dodecahedron is a rhombus with diagonal ratio 3.

The square root of 3 can be found as the leg length of an equilateral triangle that encompasses a circle with a diameter of 1.

If an equilateral triangle with sides of length 1 is cut into two equal halves, by bisecting an internal angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one and the sides are of length 1/2 and 3/2. From this the trigonometric function tangent of 60° equals 3, and the sine of 60° and the cosine of 30° both equal 3/2.

The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including[3] the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.

It is the distance between parallel sides of a regular hexagon with sides of length 1.

It is the length of the space diagonal of a unit cube.

The vesica piscis has a major axis to minor axis ratio equal to 1:3, this can be shown by constructing two equilateral triangles within it.

Other usesEdit

Power engineeringEdit

In power engineering, the voltage between two phases in a three-phase system equals 3 times the line to neutral voltage. This is because any two phases are 120° apart, and two points on a circle 120 degrees apart are separated by 3 times the radius (see geometry examples above).

See alsoEdit


  1. ^ Łukasz Komsta. "Computations | Łukasz Komsta". komsta.net. Retrieved September 24, 2016.
  2. ^ Knorr, Wilbur R. (1976), "Archimedes and the measurement of the circle: a new interpretation", Archive for History of Exact Sciences, 15 (2): 115–140, doi:10.1007/bf00348496, JSTOR 41133444, MR 0497462, S2CID 120954547.
  3. ^ Julian D. A. Wiseman Sin and Cos in Surds


  • S., D.; Jones, M. F. (1968). "22900D approximations to the square roots of the primes less than 100". Mathematics of Computation. 22 (101): 234–235. doi:10.2307/2004806. JSTOR 2004806.
  • Uhler, H. S. (1951). "Approximations exceeding 1300 decimals for  ,  ,   and distribution of digits in them". Proc. Natl. Acad. Sci. U.S.A. 37 (7): 443–447. doi:10.1073/pnas.37.7.443. PMC 1063398. PMID 16578382.
  • Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (Revised ed.). London: Penguin Group. p. 23.

External linksEdit

  • Theodorus' Constant at MathWorld
  • [1] Kevin Brown
  • [2] E. B. Davis