Square Root of 251

√251 at a glance

Exact value

√251

Decimal (10 places)

15.8429795178

Rounded

15.8 · 15.84 · 15.843

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.842980

Prime factorization

251

Cube root

6.307994

How to simplify √251

251 is a prime number, so its only factors are 1 and 251. There is no perfect-square factor to pull out, which means √251 is already in its simplest radical form.

The square root of any prime is irrational. If √251 were a fraction a/b in lowest terms, then a² = 251b², so 251 would divide a — and then 251 would divide b too, contradicting “lowest terms.” That is why the decimal 15.8429795178 is only a rounded value.

Where √251 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √251 lies between 15 and 16. 251 is 26 above 225 and 5 below 256, so the root is closer to 16.

• Straight line between 225 and 256: 15.8387 (0.03% low)
• Tangent from 15, i.e. 15 + 26 ÷ 30: 15.8667 (0.15% high)
• Tangent from 16, i.e. 16 − 5 ÷ 32: 15.8438 (0% high)

For √251 the tangent at 16 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 251 is just 5 below 256.

Finding √251 with the Babylonian method

If a guess is too big, 251 ÷ 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√251) in one step.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √251 = 15.8429795178 to every decimal shown.

√251 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √251 the pattern is [15; 1, 5, 2, 1, 2, 2, 15, 2, 2, 1, 2, 5, …] 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 √251 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 − 251y² = 1. Its smallest solution in positive whole numbers is x = 3,674,890, y = 231,957.

√251 in geometry and everyday measurements

• A square patio or deck of 251 square feet is about 15.84 ft (15 ft 10 in) on each side, so edging all the way around takes 4 × √251 ≈ 63.4 ft.
• 251 is not a sum of two whole-number squares — 251 is itself a prime that is one less than a multiple of 4, which rules that out — so √251 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 15 box, because 1² + 5² + 15² = 251.

Square roots near √251 and related roots

• The cube root of 251 is about 6.307994.
• Squaring undoes the root: (√251)² = 251, while 251² = 63,001 — the number whose square root is 251.

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