Square Root of 941

√941 at a glance

Exact value

√941

Decimal (10 places)

30.6757233004

Rounded

30.7 · 30.68 · 30.676

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.675723

Prime factorization

941

Cube root

9.799334

How to simplify √941

941 is a prime number, so its only factors are 1 and 941. There is no perfect-square factor to pull out, which means √941 is already in its simplest radical form.

The square root of any prime is irrational. If √941 were a fraction a/b in lowest terms, then a² = 941b², so 941 would divide a — and then 941 would divide b too, contradicting “lowest terms.” That is why the decimal 30.6757233004 is only a rounded value.

Where √941 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √941 lies between 30 and 31. 941 is 41 above 900 and 20 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.6721 (0.01% low)
• Tangent from 30, i.e. 30 + 41 ÷ 60: 30.6833 (0.02% high)
• Tangent from 31, i.e. 31 − 20 ÷ 62: 30.6774 (0.01% high)

For √941 the tangent at 31 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 941 is just 20 below 961.

Finding √941 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 941: following the tangent line down to zero simplifies to averaging x with 941 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √941 = 30.6757233004 to every decimal shown.

√941 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √941 the pattern is [30; 1, 2, 11, 1, 14, 2, 2, 1, 1, 2, 2, 14, …] with the block of 17 terms after the semicolon repeating forever (only the first 12 of the 17 are shown). A pattern that never ends is one more proof that √941 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 − 941y² = 1. Its smallest solution in positive whole numbers is x = 1,068,924,905,989,944,201, y = 34,845,956,052,079,180 — 19 digits for x, even though 941 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 731,069,390² − 941 × 23,832,181² = −1.

√941 in geometry and everyday measurements

• 941 square feet is 87.4 m². Laid out as a square — a small house footprint or a lot — it is about 30.68 ft (30 ft 8 in) on a side.
• 941 = 10² + 29², so by the Pythagorean theorem √941 is the diagonal of a 10 × 29 rectangle — and the distance between the points (0, 0) and (10, 29) on a grid.

Square roots near √941 and related roots

• The cube root of 941 is about 9.799334.
• Squaring undoes the root: (√941)² = 941, while 941² = 885,481 — the number whose square root is 941.

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