Square Root of 207

√207 at a glance

Exact value

3√23

Decimal (10 places)

14.3874945699

Rounded

14.4 · 14.39 · 14.387

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.387495

Prime factorization

3² × 23

Cube root

5.915482

How to simplify √207

Look for the largest perfect square that divides 207. Here it is 9 (3²), because 207 = 9 × 23 and 23 has no square factor left:

The prime factorization tells the same story: 207 = 3² × 23. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 23 stays inside.

Check: (3√23)² = 3² × 23 = 9 × 23 = 207. As a decimal, 3√23 = 3 × 4.7958315233 ≈ 14.3874945699.

Where √207 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √207 lies between 14 and 15. 207 is 11 above 196 and 18 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.3793 (0.06% low)
• Tangent from 14, i.e. 14 + 11 ÷ 28: 14.3929 (0.04% high)
• Tangent from 15, i.e. 15 − 18 ÷ 30: 14.4000 (0.09% high)

For √207 the tangent at 14 wins, missing by only 0.0054. Tangent estimates shine when the number sits close to a perfect square — here 207 is just 11 above 196.

Finding √207 with the Babylonian method

If a guess is too big, 207 ÷ 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√207) in one step.

Start from the nearest whole number, 14 (14² = 196):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √207 = 14.3874945699 to every decimal shown.

√207 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √207 the pattern is [14; 2, 1, 1, 2, 1, 1, 2, 28] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √207 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 − 207y² = 1. Its smallest solution in positive whole numbers is x = 1,151, y = 80.

√207 in geometry and everyday measurements

• A square patio or deck of 207 square feet is about 14.39 ft (14 ft 5 in) on each side, so edging all the way around takes 4 × √207 ≈ 57.5 ft.
• 207 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √207 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 √207 as its space diagonal.
• Since √207 = 3√23, a length of √207 is exactly 3 copies of the length √23 laid end to end.

Square roots near √207 and related roots

• The cube root of 207 is about 5.915482.
• Four times the radicand doubles the root: √828 = 2 × √207 ≈ 28.774989.

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