About Square Root of 3
The square root of 3 is the number that multiplied by itself gives 3. This page opens on it with 30 decimal places, computed exactly. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
To 30 places, √3 = 1.732050807568877293527446341506. Like √2 it is irrational, and 3 has no square factor, so √3 cannot be simplified further.
It appears in geometry as the height of an equilateral triangle with side 2, which is √3, so a triangle with side 1 has height √3/2, about 0.866. It is also the length of the main diagonal of a cube with side 1.
In trigonometry, tan 60° = √3 and sin 60° = √3/2, which makes it one of the exact values worth memorising.
How to use Square Root of 3
- Choose Square root, Cube root, nth root or Simplify radical.
- Type the number. A whole number, a decimal or a fraction all work, and negatives are allowed.
- For an nth root, type the root index. Choose how many decimal places you want.
- Read the exact form and the decimal at the top.
- Open the steps to see the prime factorization and the pairs taken out of the root.
Features
- Simplified radical form with the prime factorization shown: √72 = 6√2.
- Decimal to up to 100 places, computed exactly with integers.
- Square roots, cube roots and nth roots up to index 100.
- Imaginary results for negative numbers under even roots, and real roots for odd ones.
- Perfect squares and perfect powers are recognised and shown exactly.
- Decimals and fractions accepted, with the denominator rationalised.
- Step-by-step working in KaTeX and a copy button.
Facts about √3
The square of √3 is exactly 3.
(√3)² = 3
√3 ≈ 1.7320508075688772
sin 60° = √3 / 2Worked examples
√3 to 10 places gives 1.7320508076
Rounded exactly, not truncated.
√27 gives 3√3
27 = 3² × 3, so √27 = 3√3.
Privacy: how it works
- Everything is calculated by code running in your browser tab. The numbers you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- To estimate a square root by hand, find the two perfect squares around it. √50 is between √49 = 7 and √64 = 8, so it is a little over 7.
- A decimal that neither ends nor repeats means the root is irrational, which is true of the root of any number that is not a perfect power.
- Check a decimal root by squaring it. The result should be very close to the number you started with.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is the square root of 3?
About 1.7320508. It is irrational, so its decimal never ends or repeats.
Where does the square root of 3 appear?
In the height of an equilateral triangle, the diagonal of a unit cube, and the trigonometric values tan 60° and sin 60°.
More Square Root Calculator pages
- Square Root CalculatorEnter a number to get its square root in simplest radical form and as a decimal, with the working.
- Cube Root CalculatorEnter a number to get its cube root, exact where possible, with the working.
- Nth Root CalculatorEnter a number and a root index to get the nth root with the working.
- Simplify Radical CalculatorEnter the number under the root and get the radical in simplest form with the steps.
- Square Root of 2The square root of 2 is about 1.41421356, and its digits never end or repeat.
- Square Root of 72The square root of 72 simplifies to 6√2, which is about 8.4853.