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 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.
The fraction 97/56 (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 1/10,000 (approximately 9.2×10−5, with a relative error of 5×10−5). 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 1×10−11.
Archimedes reported a range for its value: (1351/780)2
> 3 > (265/153)2
. The lower limit 1351/780 is an accurate approximation for √ to 1/608400 (six decimal places, relative error 3×10−7) and the upper limit 265/153 to 2/23409 (four decimal places, relative error 1×10−5).
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.
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 √/2. From this the trigonometric function tangent of 60° equals √, and the sine of 60° and the cosine of 30° both equal √/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.
The vesica piscis has a major axis to minor axis ratio equal to 1:√, this can be shown by constructing two equilateral triangles within it.
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).