Square Root of 3

√3 at a glance

Exact value

√3

Decimal (10 places)

1.7320508076

Rounded

1.7 · 1.73 · 1.732

Perfect square?

No — between 1² and 2²

Rational?

Irrational

Both square roots

±1.732051

Prime factorization

3

Cube root

1.442250

How to simplify √3

3 is a prime number, so its only factors are 1 and 3. There is no perfect-square factor to pull out, which means √3 is already in its simplest radical form.

The square root of any prime is irrational. If √3 were a fraction a/b in lowest terms, then a² = 3b², so 3 would divide a — and then 3 would divide b too, contradicting “lowest terms.” That is why the decimal 1.7320508076 is only a rounded value.

Where √3 sits between perfect squares

1 = 1² and 4 = 2² are the nearest perfect squares, so √3 lies between 1 and 2. 3 is 2 above 1 and 1 below 4, so the root is closer to 2.

• Straight line between 1 and 4: 1.6667 (3.77% low)
• Tangent from 1, i.e. 1 + 2 ÷ 2: 2.0000 (15.47% high)
• Tangent from 2, i.e. 2 − 1 ÷ 4: 1.7500 (1.04% high)

For √3 the tangent at 2 wins, missing by only 0.0179. Tangent estimates shine when the number sits close to a perfect square — here 3 is just 1 below 4.

Finding √3 with the Babylonian method

If a guess is too big, 3 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√3) in one step.

Start from the nearest whole number, 2 (2² = 4):

The count of correct decimals went 1, 4, 8 and all 10 over 4 steps — roughly doubling each time — until the guess matched √3 = 1.7320508076 to every decimal shown.

√3 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √3 the pattern is [1; 1, 2] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √3 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 3y² = 1. Its smallest solution in positive whole numbers is x = 2, y = 1.

√3 in geometry and everyday measurements

• A square tile with an area of 3 square inches has sides about 1.732 in long.
• 3 is not a sum of two whole-number squares — 3 is itself a prime that is one less than a multiple of 4, which rules that out — so √3 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 1 box, because 1² + 1² + 1² = 3.

√3 is the space diagonal of a 1 × 1 × 1 cube, and an equilateral triangle with sides of 2 has a height of exactly √3 — the reason √3 appears in the exact values of sin 60° and tan 60°.

Square roots near √3 and related roots

• The cube root of 3 is about 1.442250.
• Multiplying the radicand by 100 multiplies the root by 10: √300 = 10 × √3 ≈ 17.320508.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.