Square Root of 675

√675 at a glance

Exact value

15√3

Decimal (10 places)

25.9807621135

Rounded

26.0 · 25.98 · 25.981

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.980762

Prime factorization

3³ × 5²

Cube root

8.772053

How to simplify √675

Look for the largest perfect square that divides 675. Here it is 225 (15²), because 675 = 225 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 675 = 3³ × 5². Each pair of equal primes leaves the radical as one factor, so 3 × 5 comes out and 3 stays inside.

675 has 3 square factors (9, 25 and 225). Starting with a smaller one still works but takes more rounds: √675 = 3√75, and √75 can be simplified again. Using 225 straight away finishes in one step.

Check: (15√3)² = 15² × 3 = 225 × 3 = 675. As a decimal, 15√3 = 15 × 1.7320508076 ≈ 25.9807621135.

Where √675 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √675 lies between 25 and 26. 675 is 50 above 625 and 1 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.9804 (0% low)
• Tangent from 25, i.e. 25 + 50 ÷ 50: 26.0000 (0.07% high)
• Tangent from 26, i.e. 26 − 1 ÷ 52: 25.9808 (0% high)

For √675 the tangent at 26 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 675 is just 1 below 676.

Finding √675 with the Babylonian method

If a guess is too big, 675 ÷ 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√675) in one step.

Start from the nearest whole number, 26 (26² = 676):

Because the starting guess was already close, two steps are enough to match √675 = 25.9807621135 to every decimal shown.

√675 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √675 the pattern is [25; 1, 50] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √675 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 − 675y² = 1. Its smallest solution in positive whole numbers is x = 26, y = 1.

√675 in geometry and everyday measurements

• 675 square feet is 62.7 m². Laid out as a square — a small house footprint or a lot — it is about 25.98 ft (26 ft) on a side.
• 675 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 √675 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 25 box, because 1² + 7² + 25² = 675.
• Since √675 = 15√3, a length of √675 is exactly 15 copies of the length √3 laid end to end.

Square roots near √675 and related roots

• The cube root of 675 is about 8.772053.
• Squaring undoes the root: (√675)² = 675, while 675² = 455,625 — the number whose square root is 675.

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