Square Root of 621

√621 at a glance

Exact value

3√69

Decimal (10 places)

24.9198715888

Rounded

24.9 · 24.92 · 24.920

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.919872

Prime factorization

3³ × 23

Cube root

8.531601

How to simplify √621

Look for the largest perfect square that divides 621. Here it is 9 (3²), because 621 = 9 × 69 and 69 has no square factor left:

The prime factorization tells the same story: 621 = 3³ × 23. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 × 23 stays inside.

Check: (3√69)² = 3² × 69 = 9 × 69 = 621. As a decimal, 3√69 = 3 × 8.3066238629 ≈ 24.9198715888.

Where √621 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √621 lies between 24 and 25. 621 is 45 above 576 and 4 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.9184 (0.01% low)
• Tangent from 24, i.e. 24 + 45 ÷ 48: 24.9375 (0.07% high)
• Tangent from 25, i.e. 25 − 4 ÷ 50: 24.9200 (0% high)

For √621 the tangent at 25 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 621 is just 4 below 625.

Finding √621 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 621: following the tangent line down to zero simplifies to averaging x with 621 ÷ x.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √621 = 24.9198715888 to every decimal shown.

√621 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √621 the pattern is [24; 1, 11, 2, 11, 1, 48] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √621 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 − 621y² = 1. Its smallest solution in positive whole numbers is x = 7,775, y = 312.

√621 in geometry and everyday measurements

• A square garage floor of 621 square feet measures about 24.92 ft (24 ft 11 in) per side, and its corner-to-corner diagonal is √1242 ≈ 35.2 ft.
• 621 is not a sum of two whole-number squares — the prime factor 3 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √621 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 16 × 19 box, because 2² + 16² + 19² = 621.
• Since √621 = 3√69, a length of √621 is exactly 3 copies of the length √69 laid end to end.

Square roots near √621 and related roots

• The cube root of 621 is about 8.531601.
• Squaring undoes the root: (√621)² = 621, while 621² = 385,641 — the number whose square root is 621.

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