Square Root of 140

√140 at a glance

Exact value

2√35

Decimal (10 places)

11.8321595662

Rounded

11.8 · 11.83 · 11.832

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.832160

Prime factorization

2² × 5 × 7

Cube root

5.192494

How to simplify √140

Look for the largest perfect square that divides 140. Here it is 4 (2²), because 140 = 4 × 35 and 35 has no square factor left:

The prime factorization tells the same story: 140 = 2² × 5 × 7. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 7 stays inside.

Check: (2√35)² = 2² × 35 = 4 × 35 = 140. As a decimal, 2√35 = 2 × 5.9160797831 ≈ 11.8321595662.

Where √140 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √140 lies between 11 and 12. 140 is 19 above 121 and 4 below 144, so the root is closer to 12.

• Straight line between 121 and 144: 11.8261 (0.05% low)
• Tangent from 11, i.e. 11 + 19 ÷ 22: 11.8636 (0.27% high)
• Tangent from 12, i.e. 12 − 4 ÷ 24: 11.8333 (0.01% high)

For √140 the tangent at 12 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 140 is just 4 below 144.

Finding √140 with the Babylonian method

Picture a rectangle with an area of 140 and one side x; the other side must be 140 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √140.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √140 = 11.8321595662 to every decimal shown.

√140 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √140 the pattern is [11; 1, 4, 1, 22] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √140 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 − 140y² = 1. Its smallest solution in positive whole numbers is x = 71, y = 6.

√140 in geometry and everyday measurements

• A square room or garden bed covering 140 square feet measures about 11.83 ft (11 ft 10 in) along each wall.
• 140 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √140 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 10 box, because 2² + 6² + 10² = 140.
• Since √140 = 2√35, a length of √140 is exactly 2 copies of the length √35 laid end to end.

Square roots near √140 and related roots

• The cube root of 140 is about 5.192494.
• Four times the radicand doubles the root: √560 = 2 × √140 ≈ 23.664319.

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