Square Root of 960

√960 at a glance

Exact value

8√15

Decimal (10 places)

30.9838667697

Rounded

31.0 · 30.98 · 30.984

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.983867

Prime factorization

2⁶ × 3 × 5

Cube root

9.864848

How to simplify √960

Look for the largest perfect square that divides 960. Here it is 64 (8²), because 960 = 64 × 15 and 15 has no square factor left:

The prime factorization tells the same story: 960 = 2⁶ × 3 × 5. Each pair of equal primes leaves the radical as one factor, so 2³ comes out and 3 × 5 stays inside.

960 has 3 square factors (4, 16 and 64). Starting with a smaller one still works but takes more rounds: √960 = 2√240, and √240 can be simplified again. Using 64 straight away finishes in one step.

Check: (8√15)² = 8² × 15 = 64 × 15 = 960. As a decimal, 8√15 = 8 × 3.8729833462 ≈ 30.9838667697.

Where √960 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √960 lies between 30 and 31. 960 is 60 above 900 and 1 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.9836 (0% low)
• Tangent from 30, i.e. 30 + 60 ÷ 60: 31.0000 (0.05% high)
• Tangent from 31, i.e. 31 − 1 ÷ 62: 30.9839 (0% high)

For √960 the tangent at 31 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 960 is just 1 below 961.

Finding √960 with the Babylonian method

Picture a rectangle with an area of 960 and one side x; the other side must be 960 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √960.

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

Because the starting guess was already close, two steps are enough to match √960 = 30.9838667697 to every decimal shown.

√960 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √960 the pattern is [30; 1, 60] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √960 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 − 960y² = 1. Its smallest solution in positive whole numbers is x = 31, y = 1.

√960 in geometry and everyday measurements

• 960 square feet is 89.2 m². Laid out as a square — a small house footprint or a lot — it is about 30.98 ft (31 ft) on a side.
• 960 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √960 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 √960 as its space diagonal.
• Since √960 = 8√15, a length of √960 is exactly 8 copies of the length √15 laid end to end.

Square roots near √960 and related roots

• The cube root of 960 is about 9.864848.
• Because 960 = 4 × 240, the root is twice √240: 2 × 15.491933 ≈ 30.983867.

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