Square Root of 940

√940 at a glance

Exact value

2√235

Decimal (10 places)

30.6594194335

Rounded

30.7 · 30.66 · 30.659

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.659419

Prime factorization

2² × 5 × 47

Cube root

9.795861

How to simplify √940

Look for the largest perfect square that divides 940. Here it is 4 (2²), because 940 = 4 × 235 and 235 has no square factor left:

The prime factorization tells the same story: 940 = 2² × 5 × 47. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 47 stays inside.

Check: (2√235)² = 2² × 235 = 4 × 235 = 940. As a decimal, 2√235 = 2 × 15.3297097168 ≈ 30.6594194335.

Where √940 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √940 lies between 30 and 31. 940 is 40 above 900 and 21 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.6557 (0.01% low)
• Tangent from 30, i.e. 30 + 40 ÷ 60: 30.6667 (0.02% high)
• Tangent from 31, i.e. 31 − 21 ÷ 62: 30.6613 (0.01% high)

For √940 the tangent at 31 wins, missing by only 0.0019. Tangent estimates shine when the number sits close to a perfect square — here 940 is just 21 below 961.

Finding √940 with the Babylonian method

Picture a rectangle with an area of 940 and one side x; the other side must be 940 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √940.

Start from the nearest whole number, 31 (31² = 961):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √940 = 30.6594194335 to every decimal shown.

√940 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √940 the pattern is [30; 1, 1, 1, 14, 1, 1, 1, 60] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √940 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 − 940y² = 1. Its smallest solution in positive whole numbers is x = 4,231, y = 138.

√940 in geometry and everyday measurements

• 940 square feet is 87.3 m². Laid out as a square — a small house footprint or a lot — it is about 30.66 ft (30 ft 8 in) on a side.
• 940 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √940 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 30 box, because 2² + 6² + 30² = 940.
• Since √940 = 2√235, a length of √940 is exactly 2 copies of the length √235 laid end to end.

Square roots near √940 and related roots

• The cube root of 940 is about 9.795861.
• Because 940 = 4 × 235, the root is twice √235: 2 × 15.32971 ≈ 30.659419.

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