Square Root of 37

√37 at a glance

Exact value

√37

Decimal (10 places)

6.0827625303

Rounded

6.1 · 6.08 · 6.083

Perfect square?

No — between 6² and 7²

Rational?

Irrational

Both square roots

±6.082763

Prime factorization

37

Cube root

3.332222

How to simplify √37

37 is a prime number, so its only factors are 1 and 37. There is no perfect-square factor to pull out, which means √37 is already in its simplest radical form.

The square root of any prime is irrational. If √37 were a fraction a/b in lowest terms, then a² = 37b², so 37 would divide a — and then 37 would divide b too, contradicting “lowest terms.” That is why the decimal 6.0827625303 is only a rounded value.

Where √37 sits between perfect squares

36 = 6² and 49 = 7² are the nearest perfect squares, so √37 lies between 6 and 7. 37 is 1 above 36 and 12 below 49, so the root is closer to 6.

• Straight line between 36 and 49: 6.0769 (0.1% low)
• Tangent from 6, i.e. 6 + 1 ÷ 12: 6.0833 (0.01% high)
• Tangent from 7, i.e. 7 − 12 ÷ 14: 6.1429 (0.99% high)

For √37 the tangent at 6 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 37 is just 1 above 36.

Finding √37 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 37: following the tangent line down to zero simplifies to averaging x with 37 ÷ x.

Start from the nearest whole number, 6 (6² = 36):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √37 = 6.0827625303 to every decimal shown.

√37 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √37 the pattern is [6; 12] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 37 is one more than a perfect square (6² + 1). A pattern that never ends is one more proof that √37 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 − 37y² = 1. Its smallest solution in positive whole numbers is x = 73, y = 12. Because the period is odd, the equation with −1 on the right also has a solution: 6² − 37 × 1² = −1.

√37 in geometry and everyday measurements

• A square room or garden bed covering 37 square feet measures about 6.08 ft (6 ft 1 in) along each wall.
• 37 = 1² + 6², so by the Pythagorean theorem √37 is the diagonal of a 1 × 6 rectangle — and the distance between the points (0, 0) and (1, 6) on a grid.

Square roots near √37 and related roots

• The cube root of 37 is about 3.332222.
• Four times the radicand doubles the root: √148 = 2 × √37 ≈ 12.165525.

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