Square Root of 585

√585 at a glance

Exact value

3√65

Decimal (10 places)

24.1867732449

Rounded

24.2 · 24.19 · 24.187

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.186773

Prime factorization

3² × 5 × 13

Cube root

8.363447

How to simplify √585

Look for the largest perfect square that divides 585. Here it is 9 (3²), because 585 = 9 × 65 and 65 has no square factor left:

The prime factorization tells the same story: 585 = 3² × 5 × 13. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 5 × 13 stays inside.

Check: (3√65)² = 3² × 65 = 9 × 65 = 585. As a decimal, 3√65 = 3 × 8.0622577483 ≈ 24.1867732449.

Where √585 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √585 lies between 24 and 25. 585 is 9 above 576 and 40 below 625, so the root is closer to 24.

• Straight line between 576 and 625: 24.1837 (0.01% low)
• Tangent from 24, i.e. 24 + 9 ÷ 48: 24.1875 (0% high)
• Tangent from 25, i.e. 25 − 40 ÷ 50: 24.2000 (0.05% high)

For √585 the tangent at 24 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 585 is just 9 above 576.

Finding √585 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 585: following the tangent line down to zero simplifies to averaging x with 585 ÷ x.

Start from the nearest whole number, 24 (24² = 576):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √585 = 24.1867732449 to every decimal shown.

√585 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √585 the pattern is [24; 5, 2, 1, 4, 1, 2, 5, 48] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √585 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 − 585y² = 1. Its smallest solution in positive whole numbers is x = 33,281, y = 1,376.

√585 in geometry and everyday measurements

• A square garage floor of 585 square feet measures about 24.19 ft (24 ft 2 in) per side, and its corner-to-corner diagonal is √1170 ≈ 34.2 ft.
• 585 = 3² + 24² = 12² + 21², so by the Pythagorean theorem √585 is the diagonal of rectangles measuring 3 × 24 and 12 × 21 — and the distance between the points (0, 0) and (3, 24) on a grid.
• Since √585 = 3√65, a length of √585 is exactly 3 copies of the length √65 laid end to end.

Square roots near √585 and related roots

• The cube root of 585 is about 8.363447.
• Squaring undoes the root: (√585)² = 585, while 585² = 342,225 — the number whose square root is 585.

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