Square Root of 263

√263 at a glance

Exact value

√263

Decimal (10 places)

16.2172747402

Rounded

16.2 · 16.22 · 16.217

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.217275

Prime factorization

263

Cube root

6.406959

How to simplify √263

263 is a prime number, so its only factors are 1 and 263. There is no perfect-square factor to pull out, which means √263 is already in its simplest radical form.

The square root of any prime is irrational. If √263 were a fraction a/b in lowest terms, then a² = 263b², so 263 would divide a — and then 263 would divide b too, contradicting “lowest terms.” That is why the decimal 16.2172747402 is only a rounded value.

Where √263 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √263 lies between 16 and 17. 263 is 7 above 256 and 26 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.2121 (0.03% low)
• Tangent from 16, i.e. 16 + 7 ÷ 32: 16.2188 (0.01% high)
• Tangent from 17, i.e. 17 − 26 ÷ 34: 16.2353 (0.11% high)

For √263 the tangent at 16 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 263 is just 7 above 256.

Finding √263 with the Babylonian method

If a guess is too big, 263 ÷ 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√263) in one step.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √263 = 16.2172747402 to every decimal shown.

√263 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √263 the pattern is [16; 4, 1, 1, 1, 1, 15, 1, 1, 1, 1, 4, 32] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √263 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 − 263y² = 1. Its smallest solution in positive whole numbers is x = 139,128, y = 8,579.

√263 in geometry and everyday measurements

• A square patio or deck of 263 square feet is about 16.22 ft (16 ft 3 in) on each side, so edging all the way around takes 4 × √263 ≈ 64.9 ft.
• 263 is not a sum of two whole-number squares — 263 is itself a prime that is one less than a multiple of 4, which rules that out — so √263 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √263 as its space diagonal.

Square roots near √263 and related roots

• The cube root of 263 is about 6.406959.
• Squaring undoes the root: (√263)² = 263, while 263² = 69,169 — the number whose square root is 263.

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