Square Root of 307

√307 at a glance

Exact value

√307

Decimal (10 places)

17.5214154679

Rounded

17.5 · 17.52 · 17.521

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.521415

Prime factorization

307

Cube root

6.745997

How to simplify √307

307 is a prime number, so its only factors are 1 and 307. There is no perfect-square factor to pull out, which means √307 is already in its simplest radical form.

The square root of any prime is irrational. If √307 were a fraction a/b in lowest terms, then a² = 307b², so 307 would divide a — and then 307 would divide b too, contradicting “lowest terms.” That is why the decimal 17.5214154679 is only a rounded value.

Where √307 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √307 lies between 17 and 18. 307 is 18 above 289 and 17 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.5143 (0.04% low)
• Tangent from 17, i.e. 17 + 18 ÷ 34: 17.5294 (0.05% high)
• Tangent from 18, i.e. 18 − 17 ÷ 36: 17.5278 (0.04% high)

For √307 the tangent at 18 wins, missing by only 0.0064. Tangent estimates shine when the number sits close to a perfect square — here 307 is just 17 below 324.

Finding √307 with the Babylonian method

If a guess is too big, 307 ÷ 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√307) in one step.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √307 = 17.5214154679 to every decimal shown.

√307 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √307 the pattern is [17; 1, 1, 11, 5, 1, 3, 17, 3, 1, 5, 11, 1, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √307 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 − 307y² = 1. Its smallest solution in positive whole numbers is x = 88,529,282, y = 5,052,633.

√307 in geometry and everyday measurements

• A square patio or deck of 307 square feet is about 17.52 ft (17 ft 6 in) on each side, so edging all the way around takes 4 × √307 ≈ 70.1 ft.
• 307 is not a sum of two whole-number squares — 307 is itself a prime that is one less than a multiple of 4, which rules that out — so √307 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 15 box, because 1² + 9² + 15² = 307.

Square roots near √307 and related roots

• The cube root of 307 is about 6.745997.
• Squaring undoes the root: (√307)² = 307, while 307² = 94,249 — the number whose square root is 307.

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