Square Root of 180

√180 at a glance

Exact value

6√5

Decimal (10 places)

13.4164078650

Rounded

13.4 · 13.42 · 13.416

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.416408

Prime factorization

2² × 3² × 5

Cube root

5.646216

How to simplify √180

Look for the largest perfect square that divides 180. Here it is 36 (6²), because 180 = 36 × 5 and 5 has no square factor left:

The prime factorization tells the same story: 180 = 2² × 3² × 5. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 5 stays inside.

180 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √180 = 2√45, and √45 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√5)² = 6² × 5 = 36 × 5 = 180. As a decimal, 6√5 = 6 × 2.2360679775 ≈ 13.4164078650.

Where √180 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √180 lies between 13 and 14. 180 is 11 above 169 and 16 below 196, so the root is closer to 13.

• Straight line between 169 and 196: 13.4074 (0.07% low)
• Tangent from 13, i.e. 13 + 11 ÷ 26: 13.4231 (0.05% high)
• Tangent from 14, i.e. 14 − 16 ÷ 28: 13.4286 (0.09% high)

For √180 the tangent at 13 wins, missing by only 0.0067. Tangent estimates shine when the number sits close to a perfect square — here 180 is just 11 above 169.

Finding √180 with the Babylonian method

Picture a rectangle with an area of 180 and one side x; the other side must be 180 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √180.

Start from the nearest whole number, 13 (13² = 169):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √180 = 13.4164078650 to every decimal shown.

√180 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √180 the pattern is [13; 2, 2, 2, 26] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √180 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 − 180y² = 1. Its smallest solution in positive whole numbers is x = 161, y = 12.

√180 in geometry and everyday measurements

• A square room or garden bed covering 180 square feet measures about 13.42 ft (13 ft 5 in) along each wall.
• 180 = 6² + 12², so by the Pythagorean theorem √180 is the diagonal of a 6 × 12 rectangle — and the distance between the points (0, 0) and (6, 12) on a grid.
• Since √180 = 6√5, a length of √180 is exactly 6 copies of the length √5 laid end to end.

Square roots near √180 and related roots

• The cube root of 180 is about 5.646216.
• Four times the radicand doubles the root: √720 = 2 × √180 ≈ 26.832816.

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