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## Summary

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.

Representations The height of an equilateral triangle with sides of length 2 equals the square root of 3. 1.7320508075688772935... $1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ddots }}}}}}}}}}$ 1.10111011011001111010... 1.BB67AE8584CAA73B...

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

1.732050807568877293527446341505872366942805253810380628055806

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
; the lower limit accurate to 1/608400 (six decimal places) and the upper limit to 2/23409 (four decimal places).

## Expressions

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:

${\begin{bmatrix}1&2\\1&3\end{bmatrix}}^{n}={\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}$

then when $n\to \infty$  :

${\sqrt {3}}=2\cdot {\frac {a_{22}}{a_{12}}}-1$

It can also be expressed by generalized continued fractions such as

$[2;-4,-4,-4,...]=2-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-\ddots }}}}}}$

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

## Geometry and trigonometry

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 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 uses

### Power engineering

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).