Square Root of 460

√460 at a glance

Exact value

2√115

Decimal (10 places)

21.4476105895

Rounded

21.4 · 21.45 · 21.448

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.447611

Prime factorization

2² × 5 × 23

Cube root

7.719443

How to simplify √460

Look for the largest perfect square that divides 460. Here it is 4 (2²), because 460 = 4 × 115 and 115 has no square factor left:

The prime factorization tells the same story: 460 = 2² × 5 × 23. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 23 stays inside.

Check: (2√115)² = 2² × 115 = 4 × 115 = 460. As a decimal, 2√115 = 2 × 10.7238052948 ≈ 21.4476105895.

Where √460 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √460 lies between 21 and 22. 460 is 19 above 441 and 24 below 484, so the root is closer to 21.

• Straight line between 441 and 484: 21.4419 (0.03% low)
• Tangent from 21, i.e. 21 + 19 ÷ 42: 21.4524 (0.02% high)
• Tangent from 22, i.e. 22 − 24 ÷ 44: 21.4545 (0.03% high)

For √460 the tangent at 21 wins, missing by only 0.0048. Tangent estimates shine when the number sits close to a perfect square — here 460 is just 19 above 441.

Finding √460 with the Babylonian method

Picture a rectangle with an area of 460 and one side x; the other side must be 460 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √460.

Start from the nearest whole number, 21 (21² = 441):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √460 = 21.4476105895 to every decimal shown.

√460 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √460 the pattern is [21; 2, 4, 3, 1, 2, 10, 2, 1, 3, 4, 2, 42] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √460 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 − 460y² = 1. Its smallest solution in positive whole numbers is x = 2,535,751, y = 118,230.

√460 in geometry and everyday measurements

• A square garage floor of 460 square feet measures about 21.45 ft (21 ft 5 in) per side, and its corner-to-corner diagonal is √920 ≈ 30.3 ft.
• 460 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √460 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 10 × 18 box, because 6² + 10² + 18² = 460.
• Since √460 = 2√115, a length of √460 is exactly 2 copies of the length √115 laid end to end.

Square roots near √460 and related roots

• The cube root of 460 is about 7.719443.
• Because 460 = 4 × 115, the root is twice √115: 2 × 10.723805 ≈ 21.447611.

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