Square Root of 904

√904 at a glance

Exact value

2√226

Decimal (10 places)

30.0665927567

Rounded

30.1 · 30.07 · 30.067

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.066593

Prime factorization

2³ × 113

Cube root

9.669176

How to simplify √904

Look for the largest perfect square that divides 904. Here it is 4 (2²), because 904 = 4 × 226 and 226 has no square factor left:

The prime factorization tells the same story: 904 = 2³ × 113. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 113 stays inside.

Check: (2√226)² = 2² × 226 = 4 × 226 = 904. As a decimal, 2√226 = 2 × 15.0332963784 ≈ 30.0665927567.

Where √904 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √904 lies between 30 and 31. 904 is 4 above 900 and 57 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.0656 (0% low)
• Tangent from 30, i.e. 30 + 4 ÷ 60: 30.0667 (0% high)
• Tangent from 31, i.e. 31 − 57 ÷ 62: 30.0806 (0.05% high)

For √904 the tangent at 30 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 904 is just 4 above 900.

Finding √904 with the Babylonian method

Picture a rectangle with an area of 904 and one side x; the other side must be 904 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √904.

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

The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √904 = 30.0665927567 to every decimal shown.

√904 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √904 the pattern is [30; 15, 60] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √904 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 − 904y² = 1. Its smallest solution in positive whole numbers is x = 451, y = 15.

√904 in geometry and everyday measurements

• 904 square feet is 84 m². Laid out as a square — a small house footprint or a lot — it is about 30.07 ft (30 ft 1 in) on a side.
• 904 = 2² + 30², so by the Pythagorean theorem √904 is the diagonal of a 2 × 30 rectangle — and the distance between the points (0, 0) and (2, 30) on a grid.
• Since √904 = 2√226, a length of √904 is exactly 2 copies of the length √226 laid end to end.

Square roots near √904 and related roots

• The cube root of 904 is about 9.669176.
• Because 904 = 4 × 226, the root is twice √226: 2 × 15.033296 ≈ 30.066593.

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