Square Root of 405

√405 at a glance

Exact value

9√5

Decimal (10 places)

20.1246117975

Rounded

20.1 · 20.12 · 20.125

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.124612

Prime factorization

3⁴ × 5

Cube root

7.398636

How to simplify √405

Look for the largest perfect square that divides 405. Here it is 81 (9²), because 405 = 81 × 5 and 5 has no square factor left:

The prime factorization tells the same story: 405 = 3⁴ × 5. Each pair of equal primes leaves the radical as one factor, so 3² comes out and 5 stays inside.

405 has 2 square factors (9 and 81). Starting with a smaller one still works but takes more rounds: √405 = 3√45, and √45 can be simplified again. Using 81 straight away finishes in one step.

Check: (9√5)² = 9² × 5 = 81 × 5 = 405. As a decimal, 9√5 = 9 × 2.2360679775 ≈ 20.1246117975.

Where √405 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √405 lies between 20 and 21. 405 is 5 above 400 and 36 below 441, so the root is closer to 20.

• Straight line between 400 and 441: 20.1220 (0.01% low)
• Tangent from 20, i.e. 20 + 5 ÷ 40: 20.1250 (0% high)
• Tangent from 21, i.e. 21 − 36 ÷ 42: 20.1429 (0.09% high)

For √405 the tangent at 20 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 405 is just 5 above 400.

Finding √405 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 405: following the tangent line down to zero simplifies to averaging x with 405 ÷ x.

Start from the nearest whole number, 20 (20² = 400):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √405 = 20.1246117975 to every decimal shown.

√405 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √405 the pattern is [20; 8, 40] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √405 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 − 405y² = 1. Its smallest solution in positive whole numbers is x = 161, y = 8.

√405 in geometry and everyday measurements

• A square garage floor of 405 square feet measures about 20.12 ft (20 ft 1 in) per side, and its corner-to-corner diagonal is √810 ≈ 28.5 ft.
• 405 = 9² + 18², so by the Pythagorean theorem √405 is the diagonal of a 9 × 18 rectangle — and the distance between the points (0, 0) and (9, 18) on a grid.
• Since √405 = 9√5, a length of √405 is exactly 9 copies of the length √5 laid end to end.

Square roots near √405 and related roots

• The cube root of 405 is about 7.398636.
• Squaring undoes the root: (√405)² = 405, while 405² = 164,025 — the number whose square root is 405.

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