Square Root of 387

√387 at a glance

Exact value

3√43

Decimal (10 places)

19.6723155729

Rounded

19.7 · 19.67 · 19.672

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.672316

Prime factorization

3² × 43

Cube root

7.287362

How to simplify √387

Look for the largest perfect square that divides 387. Here it is 9 (3²), because 387 = 9 × 43 and 43 has no square factor left:

The prime factorization tells the same story: 387 = 3² × 43. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 43 stays inside.

Check: (3√43)² = 3² × 43 = 9 × 43 = 387. As a decimal, 3√43 = 3 × 6.5574385243 ≈ 19.6723155729.

Where √387 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √387 lies between 19 and 20. 387 is 26 above 361 and 13 below 400, so the root is closer to 20.

• Straight line between 361 and 400: 19.6667 (0.03% low)
• Tangent from 19, i.e. 19 + 26 ÷ 38: 19.6842 (0.06% high)
• Tangent from 20, i.e. 20 − 13 ÷ 40: 19.6750 (0.01% high)

For √387 the tangent at 20 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 387 is just 13 below 400.

Finding √387 with the Babylonian method

If a guess is too big, 387 ÷ 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√387) in one step.

Start from the nearest whole number, 20 (20² = 400):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √387 = 19.6723155729 to every decimal shown.

√387 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √387 the pattern is [19; 1, 2, 19, 2, 1, 38] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √387 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 − 387y² = 1. Its smallest solution in positive whole numbers is x = 3,482, y = 177.

√387 in geometry and everyday measurements

• A square patio or deck of 387 square feet is about 19.67 ft (19 ft 8 in) on each side, so edging all the way around takes 4 × √387 ≈ 78.7 ft.
• 387 is not a sum of two whole-number squares — the prime factor 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √387 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 19 box, because 1² + 5² + 19² = 387.
• Since √387 = 3√43, a length of √387 is exactly 3 copies of the length √43 laid end to end.

Square roots near √387 and related roots

• The cube root of 387 is about 7.287362.
• Squaring undoes the root: (√387)² = 387, while 387² = 149,769 — the number whose square root is 387.

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