Square Root of 487

√487 at a glance

Exact value

√487

Decimal (10 places)

22.0680764907

Rounded

22.1 · 22.07 · 22.068

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.068076

Prime factorization

487

Cube root

7.867613

How to simplify √487

487 is a prime number, so its only factors are 1 and 487. There is no perfect-square factor to pull out, which means √487 is already in its simplest radical form.

The square root of any prime is irrational. If √487 were a fraction a/b in lowest terms, then a² = 487b², so 487 would divide a — and then 487 would divide b too, contradicting “lowest terms.” That is why the decimal 22.0680764907 is only a rounded value.

Where √487 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √487 lies between 22 and 23. 487 is 3 above 484 and 42 below 529, so the root is closer to 22.

• Straight line between 484 and 529: 22.0667 (0.01% low)
• Tangent from 22, i.e. 22 + 3 ÷ 44: 22.0682 (0% high)
• Tangent from 23, i.e. 23 − 42 ÷ 46: 22.0870 (0.09% high)

For √487 the tangent at 22 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 487 is just 3 above 484.

Finding √487 with the Babylonian method

If a guess is too big, 487 ÷ 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√487) in one step.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √487 = 22.0680764907 to every decimal shown.

√487 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √487 the pattern is [22; 14, 1, 2, 4, 1, 1, 3, 2, 5, 1, 6, 1, …] with the block of 28 terms after the semicolon repeating forever (only the first 12 of the 28 are shown). A pattern that never ends is one more proof that √487 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 − 487y² = 1. Its smallest solution in positive whole numbers is x = 51,906,073,840,568, y = 2,352,088,722,477 — 14 digits for x, even though 487 is small, which is what makes Pell’s equation famous.

√487 in geometry and everyday measurements

• A square garage floor of 487 square feet measures about 22.07 ft (22 ft 1 in) per side, and its corner-to-corner diagonal is √974 ≈ 31.2 ft.
• 487 is not a sum of two whole-number squares — 487 is itself a prime that is one less than a multiple of 4, which rules that out — so √487 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 √487 as its space diagonal.

Square roots near √487 and related roots

• The cube root of 487 is about 7.867613.
• Squaring undoes the root: (√487)² = 487, while 487² = 237,169 — the number whose square root is 487.

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