Square Root of 145

√145 at a glance

Exact value

√145

Decimal (10 places)

12.0415945788

Rounded

12.0 · 12.04 · 12.042

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.041595

Prime factorization

5 × 29

Cube root

5.253588

How to simplify √145

The prime factorization of 145 is 5 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √145 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 145, 5 and 29 appear an odd number of times, so √145 is irrational and 12.0415945788 is a rounded value.

Where √145 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √145 lies between 12 and 13. 145 is 1 above 144 and 24 below 169, so the root is closer to 12.

• Straight line between 144 and 169: 12.0400 (0.01% low)
• Tangent from 12, i.e. 12 + 1 ÷ 24: 12.0417 (0% high)
• Tangent from 13, i.e. 13 − 24 ÷ 26: 12.0769 (0.29% high)

For √145 the tangent at 12 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 145 is just 1 above 144.

Finding √145 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 145: following the tangent line down to zero simplifies to averaging x with 145 ÷ x.

Start from the nearest whole number, 12 (12² = 144):

The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √145 = 12.0415945788 to every decimal shown.

√145 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √145 the pattern is [12; 24] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 145 is one more than a perfect square (12² + 1). A pattern that never ends is one more proof that √145 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 − 145y² = 1. Its smallest solution in positive whole numbers is x = 289, y = 24. Because the period is odd, the equation with −1 on the right also has a solution: 12² − 145 × 1² = −1.

√145 in geometry and everyday measurements

• A square room or garden bed covering 145 square feet measures about 12.04 ft (12 ft) along each wall.
• 145 = 1² + 12² = 8² + 9², so by the Pythagorean theorem √145 is the diagonal of rectangles measuring 1 × 12 and 8 × 9 — and the distance between the points (0, 0) and (1, 12) on a grid.

Square roots near √145 and related roots

• The cube root of 145 is about 5.253588.
• Four times the radicand doubles the root: √580 = 2 × √145 ≈ 24.083189.

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