Square Root of 132

√132 at a glance

Exact value

2√33

Decimal (10 places)

11.4891252931

Rounded

11.5 · 11.49 · 11.489

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.489125

Prime factorization

2² × 3 × 11

Cube root

5.091643

How to simplify √132

Look for the largest perfect square that divides 132. Here it is 4 (2²), because 132 = 4 × 33 and 33 has no square factor left:

The prime factorization tells the same story: 132 = 2² × 3 × 11. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 11 stays inside.

Check: (2√33)² = 2² × 33 = 4 × 33 = 132. As a decimal, 2√33 = 2 × 5.7445626465 ≈ 11.4891252931.

Where √132 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √132 lies between 11 and 12. 132 is 11 above 121 and 12 below 144, so the root is closer to 11.

• Straight line between 121 and 144: 11.4783 (0.09% low)
• Tangent from 11, i.e. 11 + 11 ÷ 22: 11.5000 (0.09% high)
• Tangent from 12, i.e. 12 − 12 ÷ 24: 11.5000 (0.09% high)

For √132 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √132 with the Babylonian method

Picture a rectangle with an area of 132 and one side x; the other side must be 132 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √132.

Start from the nearest whole number, 11 (11² = 121):

The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √132 = 11.4891252931 to every decimal shown.

√132 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √132 the pattern is [11; 2, 22] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √132 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 − 132y² = 1. Its smallest solution in positive whole numbers is x = 23, y = 2.

√132 in geometry and everyday measurements

• A square room or garden bed covering 132 square feet measures about 11.49 ft (11 ft 6 in) along each wall.
• 132 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √132 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 8 × 8 box, because 2² + 8² + 8² = 132.
• Since √132 = 2√33, a length of √132 is exactly 2 copies of the length √33 laid end to end.

Square roots near √132 and related roots

• The cube root of 132 is about 5.091643.
• Four times the radicand doubles the root: √528 = 2 × √132 ≈ 22.978251.

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