Square Root of 485

√485 at a glance

Exact value

√485

Decimal (10 places)

22.0227155455

Rounded

22.0 · 22.02 · 22.023

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.022716

Prime factorization

5 × 97

Cube root

7.856828

How to simplify √485

The prime factorization of 485 is 5 × 97. Every prime appears only once, so there is no pair to bring outside the radical — √485 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 485, 5 and 97 appear an odd number of times, so √485 is irrational and 22.0227155455 is a rounded value.

Where √485 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √485 lies between 22 and 23. 485 is 1 above 484 and 44 below 529, so the root is closer to 22.

• Straight line between 484 and 529: 22.0222 (0% low)
• Tangent from 22, i.e. 22 + 1 ÷ 44: 22.0227 (0% high)
• Tangent from 23, i.e. 23 − 44 ÷ 46: 22.0435 (0.09% high)

For √485 the tangent at 22 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 485 is just 1 above 484.

Finding √485 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 485: following the tangent line down to zero simplifies to averaging x with 485 ÷ x.

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

Because the starting guess was already close, two steps are enough to match √485 = 22.0227155455 to every decimal shown.

√485 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √485 the pattern is [22; 44] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 485 is one more than a perfect square (22² + 1). A pattern that never ends is one more proof that √485 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 − 485y² = 1. Its smallest solution in positive whole numbers is x = 969, y = 44. Because the period is odd, the equation with −1 on the right also has a solution: 22² − 485 × 1² = −1.

√485 in geometry and everyday measurements

• A square garage floor of 485 square feet measures about 22.02 ft (22 ft) per side, and its corner-to-corner diagonal is √970 ≈ 31.1 ft.
• 485 = 1² + 22² = 14² + 17², so by the Pythagorean theorem √485 is the diagonal of rectangles measuring 1 × 22 and 14 × 17 — and the distance between the points (0, 0) and (1, 22) on a grid.

Square roots near √485 and related roots

• The cube root of 485 is about 7.856828.
• Squaring undoes the root: (√485)² = 485, while 485² = 235,225 — the number whose square root is 485.

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