Square Root of 375

√375 at a glance

Exact value

5√15

Decimal (10 places)

19.3649167310

Rounded

19.4 · 19.36 · 19.365

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.364917

Prime factorization

3 × 5³

Cube root

7.211248

How to simplify √375

Look for the largest perfect square that divides 375. Here it is 25 (5²), because 375 = 25 × 15 and 15 has no square factor left:

The prime factorization tells the same story: 375 = 3 × 5³. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 3 × 5 stays inside.

Check: (5√15)² = 5² × 15 = 25 × 15 = 375. As a decimal, 5√15 = 5 × 3.8729833462 ≈ 19.3649167310.

Where √375 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √375 lies between 19 and 20. 375 is 14 above 361 and 25 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.3590 (0.03% low)
• Tangent from 19, i.e. 19 + 14 ÷ 38: 19.3684 (0.02% high)
• Tangent from 20, i.e. 20 − 25 ÷ 40: 19.3750 (0.05% high)

For √375 the tangent at 19 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 375 is just 14 above 361.

Finding √375 with the Babylonian method

If a guess is too big, 375 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√375) in one step.

Start from the nearest whole number, 19 (19² = 361):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √375 = 19.3649167310 to every decimal shown.

√375 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √375 the pattern is [19; 2, 1, 2, 1, 5, 1, 2, 1, 2, 38] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √375 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 − 375y² = 1. Its smallest solution in positive whole numbers is x = 15,124, y = 781.

√375 in geometry and everyday measurements

• A square patio or deck of 375 square feet is about 19.36 ft (19 ft 4 in) on each side, so edging all the way around takes 4 × √375 ≈ 77.5 ft.
• 375 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √375 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 √375 as its space diagonal.
• Since √375 = 5√15, a length of √375 is exactly 5 copies of the length √15 laid end to end.

Square roots near √375 and related roots

• The cube root of 375 is about 7.211248.
• Squaring undoes the root: (√375)² = 375, while 375² = 140,625 — the number whose square root is 375.

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