Square Root of 580

√580 at a glance

Exact value

2√145

Decimal (10 places)

24.0831891576

Rounded

24.1 · 24.08 · 24.083

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.083189

Prime factorization

2² × 5 × 29

Cube root

8.339551

How to simplify √580

Look for the largest perfect square that divides 580. Here it is 4 (2²), because 580 = 4 × 145 and 145 has no square factor left:

The prime factorization tells the same story: 580 = 2² × 5 × 29. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 × 29 stays inside.

Check: (2√145)² = 2² × 145 = 4 × 145 = 580. As a decimal, 2√145 = 2 × 12.0415945788 ≈ 24.0831891576.

Where √580 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √580 lies between 24 and 25. 580 is 4 above 576 and 45 below 625, so the root is closer to 24.

• Straight line between 576 and 625: 24.0816 (0.01% low)
• Tangent from 24, i.e. 24 + 4 ÷ 48: 24.0833 (0% high)
• Tangent from 25, i.e. 25 − 45 ÷ 50: 24.1000 (0.07% high)

For √580 the tangent at 24 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 580 is just 4 above 576.

Finding √580 with the Babylonian method

Picture a rectangle with an area of 580 and one side x; the other side must be 580 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √580.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √580 = 24.0831891576 to every decimal shown.

√580 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √580 the pattern is [24; 12, 48] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √580 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 − 580y² = 1. Its smallest solution in positive whole numbers is x = 289, y = 12.

√580 in geometry and everyday measurements

• A square garage floor of 580 square feet measures about 24.08 ft (24 ft 1 in) per side, and its corner-to-corner diagonal is √1160 ≈ 34.1 ft.
• 580 = 2² + 24² = 16² + 18², so by the Pythagorean theorem √580 is the diagonal of rectangles measuring 2 × 24 and 16 × 18 — and the distance between the points (0, 0) and (2, 24) on a grid.
• Since √580 = 2√145, a length of √580 is exactly 2 copies of the length √145 laid end to end.

Square roots near √580 and related roots

• The cube root of 580 is about 8.339551.
• Because 580 = 4 × 145, the root is twice √145: 2 × 12.041595 ≈ 24.083189.

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