Square root of 3

Square root of 3

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

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as ${\textstyle {\sqrt {3}}}$ or ${\displaystyle 3^{1/2}}$. 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. ^[1]

In 2013, its numerical value in decimal notation was computed to ten billion digits. ^[2] Its decimal expansion, written here to 65 decimal places, is given by OEIS:  A002194 :

1.732050807568877293527446341505872366942805253810380628055806

Archimedes reported a range for its value: ${\textstyle ({\frac {1351}{780}})^{2}>3>({\frac {265}{153}})^{2}}$. ^[3]

The upper limit ${\textstyle {\frac {1351}{780}}}$ is an accurate approximation for ${\displaystyle {\sqrt {3}}}$ to ${\textstyle {\frac {1}{608,400}}}$ (six decimal places, relative error ${\textstyle 3\times 10^{-7}}$) and the lower limit ${\textstyle {\frac {265}{153}}}$ to ${\textstyle {\frac {2}{23,409}}}$ (four decimal places, relative error ${\textstyle 1\times 10^{-5}}$).

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 space diagonal of the unit cube is √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 ${\textstyle {\frac {1}{2}}}$ and ${\textstyle {\frac {\sqrt {3}}{2}}}$. From this, ${\textstyle \tan {60^{\circ }}={\sqrt {3}}}$, ${\textstyle \sin {60^{\circ }}={\frac {\sqrt {3}}{2}}}$, and ${\textstyle \cos {30^{\circ }}={\frac {\sqrt {3}}{2}}}$.

The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including ^[4] the sines of other angles.

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 ${\displaystyle 1:{\sqrt {3}}}$. This can be shown by constructing two equilateral triangles within it.

References

1. "square root of 3". planetmath.org. Retrieved 2025-07-23.
2. Komsta, Łukasz (December 2013). "Computations | Łukasz Komsta". komsta.net. WordPress. Archived from the original on 2023-10-02. Retrieved September 24, 2016.
3. 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.
4. Wiseman, Julian D. A. (June 2008). "Sin and Cos in Surds". JDAWiseman.com. Retrieved November 15, 2022.

Further reading

• Podestá, Ricardo A. (2023). "Geometric proofs that ${\displaystyle {\sqrt {3}}}$, ${\displaystyle {\sqrt {5}}}$ and ${\displaystyle {\sqrt {7}}}$ are irrational". Mathematics Magazine. 96 (1): 34–39. arXiv: 2003.06627 . doi:10.1080/0025570X.2023.2168436. MR   4556102.
• Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (Revised ed.). London: Penguin Group. p. 23.

External links

• Theodorus' Constant at MathWorld
• Kevin Brown, Archimedes and the Square Root of 3