Square Root of 124

√124 at a glance

Exact value

2√31

Decimal (10 places)

11.1355287257

Rounded

11.1 · 11.14 · 11.136

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.135529

Prime factorization

2² × 31

Cube root

4.986631

How to simplify √124

Look for the largest perfect square that divides 124. Here it is 4 (2²), because 124 = 4 × 31 and 31 has no square factor left:

The prime factorization tells the same story: 124 = 2² × 31. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 31 stays inside.

Check: (2√31)² = 2² × 31 = 4 × 31 = 124. As a decimal, 2√31 = 2 × 5.5677643628 ≈ 11.1355287257.

Where √124 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √124 lies between 11 and 12. 124 is 3 above 121 and 20 below 144, so the root is closer to 11.

• Straight line between 121 and 144: 11.1304 (0.05% low)
• Tangent from 11, i.e. 11 + 3 ÷ 22: 11.1364 (0.01% high)
• Tangent from 12, i.e. 12 − 20 ÷ 24: 11.1667 (0.28% high)

For √124 the tangent at 11 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 124 is just 3 above 121.

Finding √124 with the Babylonian method

Picture a rectangle with an area of 124 and one side x; the other side must be 124 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √124.

Start from the nearest whole number, 11 (11² = 121):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √124 = 11.1355287257 to every decimal shown.

√124 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √124 the pattern is [11; 7, 2, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √124 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 − 124y² = 1. Its smallest solution in positive whole numbers is x = 4,620,799, y = 414,960.

√124 in geometry and everyday measurements

• A square room or garden bed covering 124 square feet measures about 11.14 ft (11 ft 2 in) along each wall.
• 124 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √124 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 √124 as its space diagonal.
• Since √124 = 2√31, a length of √124 is exactly 2 copies of the length √31 laid end to end.

Square roots near √124 and related roots

• The cube root of 124 is about 4.986631.
• Four times the radicand doubles the root: √496 = 2 × √124 ≈ 22.271057.

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