Square Root of 947

√947 at a glance

Exact value

√947

Decimal (10 places)

30.7733651069

Rounded

30.8 · 30.77 · 30.773

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.773365

Prime factorization

947

Cube root

9.820117

How to simplify √947

947 is a prime number, so its only factors are 1 and 947. There is no perfect-square factor to pull out, which means √947 is already in its simplest radical form.

The square root of any prime is irrational. If √947 were a fraction a/b in lowest terms, then a² = 947b², so 947 would divide a — and then 947 would divide b too, contradicting “lowest terms.” That is why the decimal 30.7733651069 is only a rounded value.

Where √947 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √947 lies between 30 and 31. 947 is 47 above 900 and 14 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.7705 (0.01% low)
• Tangent from 30, i.e. 30 + 47 ÷ 60: 30.7833 (0.03% high)
• Tangent from 31, i.e. 31 − 14 ÷ 62: 30.7742 (0% high)

For √947 the tangent at 31 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 947 is just 14 below 961.

Finding √947 with the Babylonian method

If a guess is too big, 947 ÷ 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√947) in one step.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √947 = 30.7733651069 to every decimal shown.

√947 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √947 the pattern is [30; 1, 3, 2, 2, 2, 1, 4, 1, 7, 1, 29, 1, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √947 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 − 947y² = 1. Its smallest solution in positive whole numbers is x = 13,509,645,362, y = 439,004,487.

√947 in geometry and everyday measurements

• 947 square feet is 88 m². Laid out as a square — a small house footprint or a lot — it is about 30.77 ft (30 ft 9 in) on a side.
• 947 is not a sum of two whole-number squares — 947 is itself a prime that is one less than a multiple of 4, which rules that out — so √947 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 9 × 29 box, because 5² + 9² + 29² = 947.

Square roots near √947 and related roots

• The cube root of 947 is about 9.820117.
• Squaring undoes the root: (√947)² = 947, while 947² = 896,809 — the number whose square root is 947.

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