Square Root of 472

√472 at a glance

Exact value

2√118

Decimal (10 places)

21.7255609824

Rounded

21.7 · 21.73 · 21.726

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.725561

Prime factorization

2³ × 59

Cube root

7.785993

How to simplify √472

Look for the largest perfect square that divides 472. Here it is 4 (2²), because 472 = 4 × 118 and 118 has no square factor left:

The prime factorization tells the same story: 472 = 2³ × 59. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 59 stays inside.

Check: (2√118)² = 2² × 118 = 4 × 118 = 472. As a decimal, 2√118 = 2 × 10.8627804912 ≈ 21.7255609824.

Where √472 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √472 lies between 21 and 22. 472 is 31 above 441 and 12 below 484, so the root is closer to 22.

• Straight line between 441 and 484: 21.7209 (0.02% low)
• Tangent from 21, i.e. 21 + 31 ÷ 42: 21.7381 (0.06% high)
• Tangent from 22, i.e. 22 − 12 ÷ 44: 21.7273 (0.01% high)

For √472 the tangent at 22 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 472 is just 12 below 484.

Finding √472 with the Babylonian method

Picture a rectangle with an area of 472 and one side x; the other side must be 472 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √472.

Start from the nearest whole number, 22 (22² = 484):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √472 = 21.7255609824 to every decimal shown.

√472 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √472 the pattern is [21; 1, 2, 1, 1, 1, 4, 5, 4, 1, 1, 1, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √472 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 − 472y² = 1. Its smallest solution in positive whole numbers is x = 306,917, y = 14,127.

√472 in geometry and everyday measurements

• A square garage floor of 472 square feet measures about 21.73 ft (21 ft 9 in) per side, and its corner-to-corner diagonal is √944 ≈ 30.7 ft.
• 472 is not a sum of two whole-number squares — the prime factor 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √472 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 12 × 18 box, because 2² + 12² + 18² = 472.
• Since √472 = 2√118, a length of √472 is exactly 2 copies of the length √118 laid end to end.

Square roots near √472 and related roots

• The cube root of 472 is about 7.785993.
• Because 472 = 4 × 118, the root is twice √118: 2 × 10.86278 ≈ 21.725561.

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