Square Root of 924

√924 at a glance

Exact value

2√231

Decimal (10 places)

30.3973683071

Rounded

30.4 · 30.40 · 30.397

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.397368

Prime factorization

2² × 3 × 7 × 11

Cube root

9.739963

How to simplify √924

Look for the largest perfect square that divides 924. Here it is 4 (2²), because 924 = 4 × 231 and 231 has no square factor left:

The prime factorization tells the same story: 924 = 2² × 3 × 7 × 11. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 7 × 11 stays inside.

Check: (2√231)² = 2² × 231 = 4 × 231 = 924. As a decimal, 2√231 = 2 × 15.1986841536 ≈ 30.3973683071.

Where √924 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √924 lies between 30 and 31. 924 is 24 above 900 and 37 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.3934 (0.01% low)
• Tangent from 30, i.e. 30 + 24 ÷ 60: 30.4000 (0.01% high)
• Tangent from 31, i.e. 31 − 37 ÷ 62: 30.4032 (0.02% high)

For √924 the tangent at 30 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 924 is just 24 above 900.

Finding √924 with the Babylonian method

Picture a rectangle with an area of 924 and one side x; the other side must be 924 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √924.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √924 = 30.3973683071 to every decimal shown.

√924 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √924 the pattern is [30; 2, 1, 1, 14, 1, 1, 2, 60] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √924 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 − 924y² = 1. Its smallest solution in positive whole numbers is x = 11,551, y = 380.

√924 in geometry and everyday measurements

• 924 square feet is 85.8 m². Laid out as a square — a small house footprint or a lot — it is about 30.4 ft (30 ft 5 in) on a side.
• 924 is not a sum of two whole-number squares — the prime factor 3, 7 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √924 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 √924 as its space diagonal.
• Since √924 = 2√231, a length of √924 is exactly 2 copies of the length √231 laid end to end.

Square roots near √924 and related roots

• The cube root of 924 is about 9.739963.
• Because 924 = 4 × 231, the root is twice √231: 2 × 15.198684 ≈ 30.397368.

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