Square Root of 612

√612 at a glance

Exact value

6√17

Decimal (10 places)

24.7386337537

Rounded

24.7 · 24.74 · 24.739

Perfect square?

No — between 24² and 25²

Rational?

Irrational

Both square roots

±24.738634

Prime factorization

2² × 3² × 17

Cube root

8.490185

How to simplify √612

Look for the largest perfect square that divides 612. Here it is 36 (6²), because 612 = 36 × 17 and 17 has no square factor left:

The prime factorization tells the same story: 612 = 2² × 3² × 17. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 17 stays inside.

612 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √612 = 2√153, and √153 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√17)² = 6² × 17 = 36 × 17 = 612. As a decimal, 6√17 = 6 × 4.1231056256 ≈ 24.7386337537.

Where √612 sits between perfect squares

576 = 24² and 625 = 25² are the nearest perfect squares, so √612 lies between 24 and 25. 612 is 36 above 576 and 13 below 625, so the root is closer to 25.

• Straight line between 576 and 625: 24.7347 (0.02% low)
• Tangent from 24, i.e. 24 + 36 ÷ 48: 24.7500 (0.05% high)
• Tangent from 25, i.e. 25 − 13 ÷ 50: 24.7400 (0.01% high)

For √612 the tangent at 25 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 612 is just 13 below 625.

Finding √612 with the Babylonian method

Picture a rectangle with an area of 612 and one side x; the other side must be 612 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √612.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √612 = 24.7386337537 to every decimal shown.

√612 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √612 the pattern is [24; 1, 2, 1, 4, 1, 2, 1, 48] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √612 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 − 612y² = 1. Its smallest solution in positive whole numbers is x = 2,177, y = 88.

√612 in geometry and everyday measurements

• A square garage floor of 612 square feet measures about 24.74 ft (24 ft 9 in) per side, and its corner-to-corner diagonal is √1224 ≈ 35 ft.
• 612 = 6² + 24², so by the Pythagorean theorem √612 is the diagonal of a 6 × 24 rectangle — and the distance between the points (0, 0) and (6, 24) on a grid.
• Since √612 = 6√17, a length of √612 is exactly 6 copies of the length √17 laid end to end.

Square roots near √612 and related roots

• The cube root of 612 is about 8.490185.
• Because 612 = 4 × 153, the root is twice √153: 2 × 12.369317 ≈ 24.738634.

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