Square Root of 920

√920 at a glance

Exact value

2√230

Decimal (10 places)

30.3315017762

Rounded

30.3 · 30.33 · 30.332

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.331502

Prime factorization

2³ × 5 × 23

Cube root

9.725888

How to simplify √920

Look for the largest perfect square that divides 920. Here it is 4 (2²), because 920 = 4 × 230 and 230 has no square factor left:

The prime factorization tells the same story: 920 = 2³ × 5 × 23. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 5 × 23 stays inside.

Check: (2√230)² = 2² × 230 = 4 × 230 = 920. As a decimal, 2√230 = 2 × 15.1657508881 ≈ 30.3315017762.

Where √920 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √920 lies between 30 and 31. 920 is 20 above 900 and 41 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.3279 (0.01% low)
• Tangent from 30, i.e. 30 + 20 ÷ 60: 30.3333 (0.01% high)
• Tangent from 31, i.e. 31 − 41 ÷ 62: 30.3387 (0.02% high)

For √920 the tangent at 30 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 920 is just 20 above 900.

Finding √920 with the Babylonian method

Picture a rectangle with an area of 920 and one side x; the other side must be 920 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √920.

Start from the nearest whole number, 30 (30² = 900):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √920 = 30.3315017762 to every decimal shown.

√920 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √920 the pattern is [30; 3, 60] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √920 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 − 920y² = 1. Its smallest solution in positive whole numbers is x = 91, y = 3.

√920 in geometry and everyday measurements

• 920 square feet is 85.5 m². Laid out as a square — a small house footprint or a lot — it is about 30.33 ft (30 ft 4 in) on a side.
• 920 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √920 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 30 box, because 2² + 4² + 30² = 920.
• Since √920 = 2√230, a length of √920 is exactly 2 copies of the length √230 laid end to end.

Square roots near √920 and related roots

• The cube root of 920 is about 9.725888.
• Because 920 = 4 × 230, the root is twice √230: 2 × 15.165751 ≈ 30.331502.

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