Square Root of 934

√934 at a glance

Exact value

√934

Decimal (10 places)

30.5614135799

Rounded

30.6 · 30.56 · 30.561

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.561414

Prime factorization

2 × 467

Cube root

9.774974

How to simplify √934

The prime factorization of 934 is 2 × 467. Every prime appears only once, so there is no pair to bring outside the radical — √934 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 934, 2 and 467 appear an odd number of times, so √934 is irrational and 30.5614135799 is a rounded value.

Where √934 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √934 lies between 30 and 31. 934 is 34 above 900 and 27 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.5574 (0.01% low)
• Tangent from 30, i.e. 30 + 34 ÷ 60: 30.5667 (0.02% high)
• Tangent from 31, i.e. 31 − 27 ÷ 62: 30.5645 (0.01% high)

For √934 the tangent at 31 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 934 is just 27 below 961.

Finding √934 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 31 (31² = 961):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √934 = 30.5614135799 to every decimal shown.

√934 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √934 the pattern is [30; 1, 1, 3, 1, 1, 3, 30, 3, 1, 1, 3, 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 √934 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 − 934y² = 1. Its smallest solution in positive whole numbers is x = 3,034,565, y = 99,294.

√934 in geometry and everyday measurements

• 934 square feet is 86.8 m². Laid out as a square — a small house footprint or a lot — it is about 30.56 ft (30 ft 7 in) on a side.
• 934 is not a sum of two whole-number squares — the prime factor 467 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √934 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 30 box, because 3² + 5² + 30² = 934.

Square roots near √934 and related roots

• The cube root of 934 is about 9.774974.
• Squaring undoes the root: (√934)² = 934, while 934² = 872,356 — the number whose square root is 934.

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