Square Root of 360

√360 at a glance

Exact value

6√10

Decimal (10 places)

18.9736659610

Rounded

19.0 · 18.97 · 18.974

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.973666

Prime factorization

2³ × 3² × 5

Cube root

7.113787

How to simplify √360

Look for the largest perfect square that divides 360. Here it is 36 (6²), because 360 = 36 × 10 and 10 has no square factor left:

The prime factorization tells the same story: 360 = 2³ × 3² × 5. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 2 × 5 stays inside.

360 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √360 = 2√90, and √90 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√10)² = 6² × 10 = 36 × 10 = 360. As a decimal, 6√10 = 6 × 3.1622776602 ≈ 18.9736659610.

Where √360 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √360 lies between 18 and 19. 360 is 36 above 324 and 1 below 361, so the root is closer to 19.

• Straight line between 324 and 361: 18.9730 (0% low)
• Tangent from 18, i.e. 18 + 36 ÷ 36: 19.0000 (0.14% high)
• Tangent from 19, i.e. 19 − 1 ÷ 38: 18.9737 (0% high)

For √360 the tangent at 19 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 360 is just 1 below 361.

Finding √360 with the Babylonian method

Picture a rectangle with an area of 360 and one side x; the other side must be 360 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √360.

Start from the nearest whole number, 19 (19² = 361):

Because the starting guess was already close, two steps are enough to match √360 = 18.9736659610 to every decimal shown.

√360 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √360 the pattern is [18; 1, 36] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √360 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 − 360y² = 1. Its smallest solution in positive whole numbers is x = 19, y = 1.

√360 in geometry and everyday measurements

• A square patio or deck of 360 square feet is about 18.97 ft (19 ft) on each side, so edging all the way around takes 4 × √360 ≈ 75.9 ft.
• 360 = 6² + 18², so by the Pythagorean theorem √360 is the diagonal of a 6 × 18 rectangle — and the distance between the points (0, 0) and (6, 18) on a grid.
• Since √360 = 6√10, a length of √360 is exactly 6 copies of the length √10 laid end to end.

Square roots near √360 and related roots

• The cube root of 360 is about 7.113787.
• Because 360 = 4 × 90, the root is twice √90: 2 × 9.486833 ≈ 18.973666.

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