Square Root of 125

√125 at a glance

Exact value

5√5

Decimal (10 places)

11.1803398875

Rounded

11.2 · 11.18 · 11.180

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.180340

Prime factorization

5³

Cube root

5

How to simplify √125

Look for the largest perfect square that divides 125. Here it is 25 (5²), because 125 = 25 × 5 and 5 has no square factor left:

The prime factorization tells the same story: 125 = 5³. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 5 stays inside.

Check: (5√5)² = 5² × 5 = 25 × 5 = 125. As a decimal, 5√5 = 5 × 2.2360679775 ≈ 11.1803398875.

Where √125 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √125 lies between 11 and 12. 125 is 4 above 121 and 19 below 144, so the root is closer to 11.

• Straight line between 121 and 144: 11.1739 (0.06% low)
• Tangent from 11, i.e. 11 + 4 ÷ 22: 11.1818 (0.01% high)
• Tangent from 12, i.e. 12 − 19 ÷ 24: 11.2083 (0.25% high)

For √125 the tangent at 11 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 125 is just 4 above 121.

Finding √125 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 125: following the tangent line down to zero simplifies to averaging x with 125 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √125 = 11.1803398875 to every decimal shown.

√125 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √125 the pattern is [11; 5, 1, 1, 5, 22] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √125 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 − 125y² = 1. Its smallest solution in positive whole numbers is x = 930,249, y = 83,204. Because the period is odd, the equation with −1 on the right also has a solution: 682² − 125 × 61² = −1.

√125 in geometry and everyday measurements

• A square room or garden bed covering 125 square feet measures about 11.18 ft (11 ft 2 in) along each wall.
• 125 = 2² + 11² = 5² + 10², so by the Pythagorean theorem √125 is the diagonal of rectangles measuring 2 × 11 and 5 × 10 — and the distance between the points (0, 0) and (2, 11) on a grid.
• Since √125 = 5√5, a length of √125 is exactly 5 copies of the length √5 laid end to end.

Square roots near √125 and related roots

• The cube root of 125 is exactly 5 — 125 is a perfect cube as well (5³).
• Four times the radicand doubles the root: √500 = 2 × √125 ≈ 22.36068.

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