Square Root of 925

√925 at a glance

Exact value

5√37

Decimal (10 places)

30.4138126515

Rounded

30.4 · 30.41 · 30.414

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.413813

Prime factorization

5² × 37

Cube root

9.743476

How to simplify √925

Look for the largest perfect square that divides 925. Here it is 25 (5²), because 925 = 25 × 37 and 37 has no square factor left:

The prime factorization tells the same story: 925 = 5² × 37. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 37 stays inside.

Check: (5√37)² = 5² × 37 = 25 × 37 = 925. As a decimal, 5√37 = 5 × 6.0827625303 ≈ 30.4138126515.

Where √925 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √925 lies between 30 and 31. 925 is 25 above 900 and 36 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.4098 (0.01% low)
• Tangent from 30, i.e. 30 + 25 ÷ 60: 30.4167 (0.01% high)
• Tangent from 31, i.e. 31 − 36 ÷ 62: 30.4194 (0.02% high)

For √925 the tangent at 30 wins, missing by only 0.0029. Tangent estimates shine when the number sits close to a perfect square — here 925 is just 25 above 900.

Finding √925 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 925: following the tangent line down to zero simplifies to averaging x with 925 ÷ x.

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 √925 = 30.4138126515 to every decimal shown.

√925 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √925 the pattern is [30; 2, 2, 2, 2, 60] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √925 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 − 925y² = 1. Its smallest solution in positive whole numbers is x = 1,555,849, y = 51,156. Because the period is odd, the equation with −1 on the right also has a solution: 882² − 925 × 29² = −1.

√925 in geometry and everyday measurements

• 925 square feet is 85.9 m². Laid out as a square — a small house footprint or a lot — it is about 30.41 ft (30 ft 5 in) on a side.
• 925 = 5² + 30² = 14² + 27² = 21² + 22², so by the Pythagorean theorem √925 is the diagonal of rectangles measuring 5 × 30, 14 × 27 and 21 × 22 — and the distance between the points (0, 0) and (5, 30) on a grid.
• Since √925 = 5√37, a length of √925 is exactly 5 copies of the length √37 laid end to end.

Square roots near √925 and related roots

• The cube root of 925 is about 9.743476.
• Squaring undoes the root: (√925)² = 925, while 925² = 855,625 — the number whose square root is 925.

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