Square Root of 962

√962 at a glance

Exact value

√962

Decimal (10 places)

31.0161248385

Rounded

31.0 · 31.02 · 31.016

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.016125

Prime factorization

2 × 13 × 37

Cube root

9.871694

How to simplify √962

The prime factorization of 962 is 2 × 13 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √962 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 962, 2, 13 and 37 appear an odd number of times, so √962 is irrational and 31.0161248385 is a rounded value.

Where √962 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √962 lies between 31 and 32. 962 is 1 above 961 and 62 below 1,024, so the root is closer to 31.

• Straight line between 961 and 1,024: 31.0159 (0% low)
• Tangent from 31, i.e. 31 + 1 ÷ 62: 31.0161 (0% high)
• Tangent from 32, i.e. 32 − 62 ÷ 64: 31.0313 (0.05% high)

For √962 the tangent at 31 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 962 is just 1 above 961.

Finding √962 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):

Because the starting guess was already close, two steps are enough to match √962 = 31.0161248385 to every decimal shown.

√962 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √962 the pattern is [31; 62] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 962 is one more than a perfect square (31² + 1). A pattern that never ends is one more proof that √962 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 − 962y² = 1. Its smallest solution in positive whole numbers is x = 1,923, y = 62. Because the period is odd, the equation with −1 on the right also has a solution: 31² − 962 × 1² = −1.

√962 in geometry and everyday measurements

• 962 square feet is 89.4 m². Laid out as a square — a small house footprint or a lot — it is about 31.02 ft (31 ft) on a side.
• 962 = 1² + 31² = 11² + 29², so by the Pythagorean theorem √962 is the diagonal of rectangles measuring 1 × 31 and 11 × 29 — and the distance between the points (0, 0) and (1, 31) on a grid.

Square roots near √962 and related roots

• The cube root of 962 is about 9.871694.
• Squaring undoes the root: (√962)² = 962, while 962² = 925,444 — the number whose square root is 962.

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