Square Root of 370

√370 at a glance

Exact value

√370

Decimal (10 places)

19.2353840617

Rounded

19.2 · 19.24 · 19.235

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.235384

Prime factorization

2 × 5 × 37

Cube root

7.179054

How to simplify √370

The prime factorization of 370 is 2 × 5 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √370 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 370, 2, 5 and 37 appear an odd number of times, so √370 is irrational and 19.2353840617 is a rounded value.

Where √370 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √370 lies between 19 and 20. 370 is 9 above 361 and 30 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.2308 (0.02% low)
• Tangent from 19, i.e. 19 + 9 ÷ 38: 19.2368 (0.01% high)
• Tangent from 20, i.e. 20 − 30 ÷ 40: 19.2500 (0.08% high)

For √370 the tangent at 19 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 370 is just 9 above 361.

Finding √370 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √370 = 19.2353840617 to every decimal shown.

√370 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √370 the pattern is [19; 4, 4, 38] with the block of 3 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √370 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 − 370y² = 1. Its smallest solution in positive whole numbers is x = 213,859, y = 11,118. Because the period is odd, the equation with −1 on the right also has a solution: 327² − 370 × 17² = −1.

√370 in geometry and everyday measurements

• A square patio or deck of 370 square feet is about 19.24 ft (19 ft 3 in) on each side, so edging all the way around takes 4 × √370 ≈ 76.9 ft.
• 370 = 3² + 19² = 9² + 17², so by the Pythagorean theorem √370 is the diagonal of rectangles measuring 3 × 19 and 9 × 17 — and the distance between the points (0, 0) and (3, 19) on a grid.

Square roots near √370 and related roots

• The cube root of 370 is about 7.179054.
• Squaring undoes the root: (√370)² = 370, while 370² = 136,900 — the number whose square root is 370.

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