Square Root of 373

√373 at a glance

Exact value

√373

Decimal (10 places)

19.3132079158

Rounded

19.3 · 19.31 · 19.313

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.313208

Prime factorization

373

Cube root

7.198405

How to simplify √373

373 is a prime number, so its only factors are 1 and 373. There is no perfect-square factor to pull out, which means √373 is already in its simplest radical form.

The square root of any prime is irrational. If √373 were a fraction a/b in lowest terms, then a² = 373b², so 373 would divide a — and then 373 would divide b too, contradicting “lowest terms.” That is why the decimal 19.3132079158 is only a rounded value.

Where √373 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √373 lies between 19 and 20. 373 is 12 above 361 and 27 below 400, so the root is closer to 19.

• Straight line between 361 and 400: 19.3077 (0.03% low)
• Tangent from 19, i.e. 19 + 12 ÷ 38: 19.3158 (0.01% high)
• Tangent from 20, i.e. 20 − 27 ÷ 40: 19.3250 (0.06% high)

For √373 the tangent at 19 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 373 is just 12 above 361.

Finding √373 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 373: following the tangent line down to zero simplifies to averaging x with 373 ÷ x.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √373 = 19.3132079158 to every decimal shown.

√373 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √373 the pattern is [19; 3, 5, 5, 3, 38] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √373 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 − 373y² = 1. Its smallest solution in positive whole numbers is x = 52,387,849, y = 2,712,540. Because the period is odd, the equation with −1 on the right also has a solution: 5,118² − 373 × 265² = −1.

√373 in geometry and everyday measurements

• A square patio or deck of 373 square feet is about 19.31 ft (19 ft 4 in) on each side, so edging all the way around takes 4 × √373 ≈ 77.3 ft.
• 373 = 7² + 18², so by the Pythagorean theorem √373 is the diagonal of a 7 × 18 rectangle — and the distance between the points (0, 0) and (7, 18) on a grid.

Square roots near √373 and related roots

• The cube root of 373 is about 7.198405.
• Squaring undoes the root: (√373)² = 373, while 373² = 139,129 — the number whose square root is 373.

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