Square Root of 233

√233 at a glance

Exact value

√233

Decimal (10 places)

15.2643375225

Rounded

15.3 · 15.26 · 15.264

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.264338

Prime factorization

233

Cube root

6.153449

How to simplify √233

233 is a prime number, so its only factors are 1 and 233. There is no perfect-square factor to pull out, which means √233 is already in its simplest radical form.

The square root of any prime is irrational. If √233 were a fraction a/b in lowest terms, then a² = 233b², so 233 would divide a — and then 233 would divide b too, contradicting “lowest terms.” That is why the decimal 15.2643375225 is only a rounded value.

Where √233 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √233 lies between 15 and 16. 233 is 8 above 225 and 23 below 256, so the root is closer to 15.

• Straight line between 225 and 256: 15.2581 (0.04% low)
• Tangent from 15, i.e. 15 + 8 ÷ 30: 15.2667 (0.02% high)
• Tangent from 16, i.e. 16 − 23 ÷ 32: 15.2813 (0.11% high)

For √233 the tangent at 15 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 233 is just 8 above 225.

Finding √233 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 233: following the tangent line down to zero simplifies to averaging x with 233 ÷ x.

Start from the nearest whole number, 15 (15² = 225):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √233 = 15.2643375225 to every decimal shown.

√233 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √233 the pattern is [15; 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 30] with the block of 11 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √233 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 − 233y² = 1. Its smallest solution in positive whole numbers is x = 1,072,400,673, y = 70,255,304. Because the period is odd, the equation with −1 on the right also has a solution: 23,156² − 233 × 1,517² = −1.

√233 in geometry and everyday measurements

• A square patio or deck of 233 square feet is about 15.26 ft (15 ft 3 in) on each side, so edging all the way around takes 4 × √233 ≈ 61.1 ft.
• 233 = 8² + 13², so by the Pythagorean theorem √233 is the diagonal of a 8 × 13 rectangle — and the distance between the points (0, 0) and (8, 13) on a grid.

Square roots near √233 and related roots

• The cube root of 233 is about 6.153449.
• Four times the radicand doubles the root: √932 = 2 × √233 ≈ 30.528675.

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