Square Root of 952

√952 at a glance

Exact value

2√238

Decimal (10 places)

30.8544972411

Rounded

30.9 · 30.85 · 30.854

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.854497

Prime factorization

2³ × 7 × 17

Cube root

9.837369

How to simplify √952

Look for the largest perfect square that divides 952. Here it is 4 (2²), because 952 = 4 × 238 and 238 has no square factor left:

The prime factorization tells the same story: 952 = 2³ × 7 × 17. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 7 × 17 stays inside.

Check: (2√238)² = 2² × 238 = 4 × 238 = 952. As a decimal, 2√238 = 2 × 15.4272486205 ≈ 30.8544972411.

Where √952 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √952 lies between 30 and 31. 952 is 52 above 900 and 9 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.8525 (0.01% low)
• Tangent from 30, i.e. 30 + 52 ÷ 60: 30.8667 (0.04% high)
• Tangent from 31, i.e. 31 − 9 ÷ 62: 30.8548 (0% high)

For √952 the tangent at 31 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 952 is just 9 below 961.

Finding √952 with the Babylonian method

Picture a rectangle with an area of 952 and one side x; the other side must be 952 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √952.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √952 = 30.8544972411 to every decimal shown.

√952 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √952 the pattern is [30; 1, 5, 1, 6, 1, 5, 1, 60] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √952 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 − 952y² = 1. Its smallest solution in positive whole numbers is x = 11,663, y = 378.

√952 in geometry and everyday measurements

• 952 square feet is 88.4 m². Laid out as a square — a small house footprint or a lot — it is about 30.85 ft (30 ft 10 in) on a side.
• 952 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √952 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 6 × 30 box, because 4² + 6² + 30² = 952.
• Since √952 = 2√238, a length of √952 is exactly 2 copies of the length √238 laid end to end.

Square roots near √952 and related roots

• The cube root of 952 is about 9.837369.
• Because 952 = 4 × 238, the root is twice √238: 2 × 15.427249 ≈ 30.854497.

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