Square Root of 325

√325 at a glance

Exact value

5√13

Decimal (10 places)

18.0277563773

Rounded

18.0 · 18.03 · 18.028

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.027756

Prime factorization

5² × 13

Cube root

6.875344

How to simplify √325

Look for the largest perfect square that divides 325. Here it is 25 (5²), because 325 = 25 × 13 and 13 has no square factor left:

The prime factorization tells the same story: 325 = 5² × 13. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 13 stays inside.

Check: (5√13)² = 5² × 13 = 25 × 13 = 325. As a decimal, 5√13 = 5 × 3.6055512755 ≈ 18.0277563773.

Where √325 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √325 lies between 18 and 19. 325 is 1 above 324 and 36 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.0270 (0% low)
• Tangent from 18, i.e. 18 + 1 ÷ 36: 18.0278 (0% high)
• Tangent from 19, i.e. 19 − 36 ÷ 38: 18.0526 (0.14% high)

For √325 the tangent at 18 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 325 is just 1 above 324.

Finding √325 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 325: following the tangent line down to zero simplifies to averaging x with 325 ÷ x.

Start from the nearest whole number, 18 (18² = 324):

Because the starting guess was already close, two steps are enough to match √325 = 18.0277563773 to every decimal shown.

√325 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √325 the pattern is [18; 36] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 325 is one more than a perfect square (18² + 1). A pattern that never ends is one more proof that √325 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 − 325y² = 1. Its smallest solution in positive whole numbers is x = 649, y = 36. Because the period is odd, the equation with −1 on the right also has a solution: 18² − 325 × 1² = −1.

√325 in geometry and everyday measurements

• A square patio or deck of 325 square feet is about 18.03 ft (18 ft) on each side, so edging all the way around takes 4 × √325 ≈ 72.1 ft.
• 325 = 1² + 18² = 6² + 17² = 10² + 15², so by the Pythagorean theorem √325 is the diagonal of rectangles measuring 1 × 18, 6 × 17 and 10 × 15 — and the distance between the points (0, 0) and (1, 18) on a grid.
• Since √325 = 5√13, a length of √325 is exactly 5 copies of the length √13 laid end to end.

Square roots near √325 and related roots

• The cube root of 325 is about 6.875344.
• Squaring undoes the root: (√325)² = 325, while 325² = 105,625 — the number whose square root is 325.

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