Square Root of 121

√121 at a glance

Exact value

11

Decimal (10 places)

11

Rounded

11

Perfect square?

Yes — 11²

Rational?

Rational (whole number)

Both square roots

±11

Prime factorization

11²

Cube root

4.946087

How to simplify √121

Write 121 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:

121 has 2 factor pairs (1 × 121, 11 × 11). The pair with two equal numbers, 11 × 11, is what makes it a perfect square — and why it has an odd number of factors.

Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 121 = 1 + 3 + 5 + 7 + … + 21, the first 11 odd numbers.

Where √121 sits between perfect squares

121 is 11². The perfect squares on either side are 100 (10²) and 144 (12²), so the gaps are 21 below and 23 above — consecutive squares always differ by consecutive odd numbers.

Finding √121 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 121: following the tangent line down to zero simplifies to averaging x with 121 ÷ x.

To show the method at work on a perfect square, start one above the answer, at 12:

The count of correct decimals went 1, 4, 9 and all 10 over 4 steps — roughly doubling each time — until the guess matched √121 = 11.0000000000 to every decimal shown.

√121 in geometry and everyday measurements

• A square room or garden bed covering 121 square feet measures exactly 11 ft (11 ft) along each wall.
• 121 is not a sum of two whole-number squares — no pair of squares adds up to it — so √121 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 9 box, because 2² + 6² + 9² = 121.

Square roots near √121 and related roots

• The cube root of 121 is about 4.946087.
• Taking the square root twice gives the fourth root: √11 ≈ 3.316625.
• Four times the radicand doubles the root: √484 = 2 × √121 ≈ 22.

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