Square root of 3

Represent root 10 ( √10 ) on number line
Represent root 10 ( √10 ) on number line

Square root of 3

The height of an equilateral triangle with sides of length 2 equals the square root of 3.

Decimal 1.7320508075688772935…
Continued fraction
Binary 1.10111011011001111010…
Hexadecimal 1.BB67AE8584CAA73B…

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as or . 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.

As of December 2013[update], 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 OEIS: A002194:


The fraction (1.732142857…) can be used as a good approximation. Despite having a denominator of only 56, it differs from the correct value by less than (approximately , with a relative error of ). The rounded value of 1.732 is correct to within 0.01% of the actual value.

The fraction (1.73205080756…) is accurate to .

Archimedes reported a range for its value: .[2]

The lower limit is an accurate approximation for to (six decimal places, relative error ) and the upper limit to (four decimal places, relative error ).


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

So it is 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 trigonometry[edit]

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 and . From this, , , and .

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 . This can be shown by constructing two equilateral triangles within it.

Other uses and occurrence[edit]

Power engineering[edit]

In power engineering, the voltage between two phases in a three-phase system equals 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 times the radius (see geometry examples above).

Special functions[edit]

It is known that most roots of the nth derivatives of (where n < 18 and is the Bessel function of the first kind of order ) are transcendental. The only exceptions are the numbers , which are the algebraic roots of both and . [4][clarification needed]

See also[edit]

Other references[edit]


  1. ^ Komsta, Łukasz (December 2013). “Computations | Łukasz Komsta”. komsta.net. WordPress. Retrieved September 24, 2016.{{cite web}}: CS1 maint: url-status (link)
  2. ^ Knorr, Wilbur R. (June 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. Retrieved November 15, 2022 – via SpringerLink.
  3. ^ Wiseman, Julian D. A. (June 2008). “Sin and Cos in Surds”. JDAWiseman.com. Retrieved November 15, 2022.{{cite web}}: CS1 maint: url-status (link)
  4. ^ Lorch, Lee; Muldoon, Martin E. (1995). “Transcendentality of zeros of higher dereivatives of functions involving Bessel functions”. International Journal of Mathematics and Mathematical Sciences. 18 (3): 551–560. doi:10.1155/S0161171295000706.
  5. ^ 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.
  6. ^ 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.
  7. ^ Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (Revised ed.). London: Penguin Group. p. 23.
  • Podestá, Ricardo A. (2020). “A geometric proof that sqrt 3, sqrt 5, and sqrt 7 are irrational”. arXiv:2003.06627 [math.GM].

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