Square Root of 467

√467 at a glance

Exact value

√467

Decimal (10 places)

21.6101827850

Rounded

21.6 · 21.61 · 21.610

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.610183

Prime factorization

467

Cube root

7.758402

How to simplify √467

467 is a prime number, so its only factors are 1 and 467. There is no perfect-square factor to pull out, which means √467 is already in its simplest radical form.

The square root of any prime is irrational. If √467 were a fraction a/b in lowest terms, then a² = 467b², so 467 would divide a — and then 467 would divide b too, contradicting “lowest terms.” That is why the decimal 21.6101827850 is only a rounded value.

Where √467 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √467 lies between 21 and 22. 467 is 26 above 441 and 17 below 484, so the root is closer to 22.

• Straight line between 441 and 484: 21.6047 (0.03% low)
• Tangent from 21, i.e. 21 + 26 ÷ 42: 21.6190 (0.04% high)
• Tangent from 22, i.e. 22 − 17 ÷ 44: 21.6136 (0.02% high)

For √467 the tangent at 22 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 467 is just 17 below 484.

Finding √467 with the Babylonian method

If a guess is too big, 467 ÷ 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√467) in one step.

Start from the nearest whole number, 22 (22² = 484):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √467 = 21.6101827850 to every decimal shown.

√467 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √467 the pattern is [21; 1, 1, 1, 1, 3, 3, 21, 3, 3, 1, 1, 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 √467 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 − 467y² = 1. Its smallest solution in positive whole numbers is x = 1,625,626, y = 75,225.

√467 in geometry and everyday measurements

• A square garage floor of 467 square feet measures about 21.61 ft (21 ft 7 in) per side, and its corner-to-corner diagonal is √934 ≈ 30.6 ft.
• 467 is not a sum of two whole-number squares — 467 is itself a prime that is one less than a multiple of 4, which rules that out — so √467 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 21 box, because 1² + 5² + 21² = 467.

Square roots near √467 and related roots

• The cube root of 467 is about 7.758402.
• Squaring undoes the root: (√467)² = 467, while 467² = 218,089 — the number whose square root is 467.

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