Square root of 3

Square root of 3

File:Equilateral triangle with side 2.svg The height of an equilateral triangle with sides of length 2 equals the square root of 3.
Representations
Decimal	1.7320508075688772935...
Continued fraction	$1+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\ddots }}}}}$

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as $\sqrt{3}$ or $3^{1/2}$. 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's constant, after Theodorus of Cyrene, who proved its irrationality.

In 2013, its numerical value in decimal notation was computed to ten billion digits. Its decimal expansion, written here to 60 decimal places, is given by OEIS: A002194:

1.732050807568877293527446341505872366942805253810380628055806

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

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

Geometry and trigonometry

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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.

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And, the height of a regular hexagon with sides of length 1

File:Square root of 3 in cube.svg

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 $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$. From this, $\tan 60^{\circ }=\sqrt{3}$, $\sin 60^{\circ }=\frac{\sqrt{3}}{2}$, and $\cos 30^{\circ }=\frac{\sqrt{3}}{2}$.

The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including 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 $\sqrt{3}:1$. This can be shown by constructing two equilateral triangles within it.

External links

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