Square Root of 945

√945 at a glance

Exact value

3√105

Decimal (10 places)

30.7408522979

Rounded

30.7 · 30.74 · 30.741

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.740852

Prime factorization

3³ × 5 × 7

Cube root

9.813199

How to simplify √945

Look for the largest perfect square that divides 945. Here it is 9 (3²), because 945 = 9 × 105 and 105 has no square factor left:

The prime factorization tells the same story: 945 = 3³ × 5 × 7. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 × 5 × 7 stays inside.

Check: (3√105)² = 3² × 105 = 9 × 105 = 945. As a decimal, 3√105 = 3 × 10.246950766 ≈ 30.7408522979.

Where √945 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √945 lies between 30 and 31. 945 is 45 above 900 and 16 below 961, so the root is closer to 31.

• Straight line between 900 and 961: 30.7377 (0.01% low)
• Tangent from 30, i.e. 30 + 45 ÷ 60: 30.7500 (0.03% high)
• Tangent from 31, i.e. 31 − 16 ÷ 62: 30.7419 (0% high)

For √945 the tangent at 31 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 945 is just 16 below 961.

Finding √945 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 945: following the tangent line down to zero simplifies to averaging x with 945 ÷ x.

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 √945 = 30.7408522979 to every decimal shown.

√945 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √945 the pattern is [30; 1, 2, 1, 6, 12, 6, 1, 2, 1, 60] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √945 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 − 945y² = 1. Its smallest solution in positive whole numbers is x = 275,561, y = 8,964.

√945 in geometry and everyday measurements

• 945 square feet is 87.8 m². Laid out as a square — a small house footprint or a lot — it is about 30.74 ft (30 ft 9 in) on a side.
• 945 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √945 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 10 × 29 box, because 2² + 10² + 29² = 945.
• Since √945 = 3√105, a length of √945 is exactly 3 copies of the length √105 laid end to end.

Square roots near √945 and related roots

• The cube root of 945 is about 9.813199.
• Squaring undoes the root: (√945)² = 945, while 945² = 893,025 — the number whose square root is 945.

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