Square Root of 956

√956 at a glance

Exact value

2√239

Decimal (10 places)

30.9192496675

Rounded

30.9 · 30.92 · 30.919

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.919250

Prime factorization

2² × 239

Cube root

9.851128

How to simplify √956

Look for the largest perfect square that divides 956. Here it is 4 (2²), because 956 = 4 × 239 and 239 has no square factor left:

The prime factorization tells the same story: 956 = 2² × 239. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 239 stays inside.

Check: (2√239)² = 2² × 239 = 4 × 239 = 956. As a decimal, 2√239 = 2 × 15.4596248337 ≈ 30.9192496675.

Where √956 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √956 lies between 30 and 31. 956 is 56 above 900 and 5 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.9180 (0% low)
• Tangent from 30, i.e. 30 + 56 ÷ 60: 30.9333 (0.05% high)
• Tangent from 31, i.e. 31 − 5 ÷ 62: 30.9194 (0% high)

For √956 the tangent at 31 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 956 is just 5 below 961.

Finding √956 with the Babylonian method

Picture a rectangle with an area of 956 and one side x; the other side must be 956 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √956.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √956 = 30.9192496675 to every decimal shown.

√956 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √956 the pattern is [30; 1, 11, 2, 1, 1, 1, 1, 7, 8, 1, 2, 2, …] with the block of 32 terms after the semicolon repeating forever (only the first 12 of the 32 are shown). A pattern that never ends is one more proof that √956 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 − 956y² = 1. Its smallest solution in positive whole numbers is x = 76,759,023,628,799, y = 2,482,564,242,480 — 14 digits for x, even though 956 is small, which is what makes Pell’s equation famous.

√956 in geometry and everyday measurements

• 956 square feet is 88.8 m². Laid out as a square — a small house footprint or a lot — it is about 30.92 ft (30 ft 11 in) on a side.
• 956 is not a sum of two whole-number squares — the prime factor 239 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √956 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 √956 as its space diagonal.
• Since √956 = 2√239, a length of √956 is exactly 2 copies of the length √239 laid end to end.

Square roots near √956 and related roots

• The cube root of 956 is about 9.851128.
• Because 956 = 4 × 239, the root is twice √239: 2 × 15.459625 ≈ 30.91925.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.