Square Root of 480

√480 at a glance

Exact value

4√30

Decimal (10 places)

21.9089023002

Rounded

21.9 · 21.91 · 21.909

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.908902

Prime factorization

2⁵ × 3 × 5

Cube root

7.829735

How to simplify √480

Look for the largest perfect square that divides 480. Here it is 16 (4²), because 480 = 16 × 30 and 30 has no square factor left:

The prime factorization tells the same story: 480 = 2⁵ × 3 × 5. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 3 × 5 stays inside.

480 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √480 = 2√120, and √120 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√30)² = 4² × 30 = 16 × 30 = 480. As a decimal, 4√30 = 4 × 5.4772255751 ≈ 21.9089023002.

Where √480 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √480 lies between 21 and 22. 480 is 39 above 441 and 4 below 484, so the root is closer to 22.

• Straight line between 441 and 484: 21.9070 (0.01% low)
• Tangent from 21, i.e. 21 + 39 ÷ 42: 21.9286 (0.09% high)
• Tangent from 22, i.e. 22 − 4 ÷ 44: 21.9091 (0% high)

For √480 the tangent at 22 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 480 is just 4 below 484.

Finding √480 with the Babylonian method

Picture a rectangle with an area of 480 and one side x; the other side must be 480 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √480.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √480 = 21.9089023002 to every decimal shown.

√480 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √480 the pattern is [21; 1, 9, 1, 42] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √480 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 − 480y² = 1. Its smallest solution in positive whole numbers is x = 241, y = 11.

√480 in geometry and everyday measurements

• A square garage floor of 480 square feet measures about 21.91 ft (21 ft 11 in) per side, and its corner-to-corner diagonal is √960 ≈ 31 ft.
• 480 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √480 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 8 × 20 box, because 4² + 8² + 20² = 480.
• Since √480 = 4√30, a length of √480 is exactly 4 copies of the length √30 laid end to end.

Square roots near √480 and related roots

• The cube root of 480 is about 7.829735.
• Because 480 = 4 × 120, the root is twice √120: 2 × 10.954451 ≈ 21.908902.

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