Square Root of 410

√410 at a glance

Exact value

√410

Decimal (10 places)

20.2484567313

Rounded

20.2 · 20.25 · 20.248

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.248457

Prime factorization

2 × 5 × 41

Cube root

7.428959

How to simplify √410

The prime factorization of 410 is 2 × 5 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √410 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 410, 2, 5 and 41 appear an odd number of times, so √410 is irrational and 20.2484567313 is a rounded value.

Where √410 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √410 lies between 20 and 21. 410 is 10 above 400 and 31 below 441, so the root is closer to 20.

• Straight line between 400 and 441: 20.2439 (0.02% low)
• Tangent from 20, i.e. 20 + 10 ÷ 40: 20.2500 (0.01% high)
• Tangent from 21, i.e. 21 − 31 ÷ 42: 20.2619 (0.07% high)

For √410 the tangent at 20 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 410 is just 10 above 400.

Finding √410 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √410 = 20.2484567313 to every decimal shown.

√410 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √410 the pattern is [20; 4, 40] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √410 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 − 410y² = 1. Its smallest solution in positive whole numbers is x = 81, y = 4.

√410 in geometry and everyday measurements

• A square garage floor of 410 square feet measures about 20.25 ft (20 ft 3 in) per side, and its corner-to-corner diagonal is √820 ≈ 28.6 ft.
• 410 = 7² + 19² = 11² + 17², so by the Pythagorean theorem √410 is the diagonal of rectangles measuring 7 × 19 and 11 × 17 — and the distance between the points (0, 0) and (7, 19) on a grid.

Square roots near √410 and related roots

• The cube root of 410 is about 7.428959.
• Squaring undoes the root: (√410)² = 410, while 410² = 168,100 — the number whose square root is 410.

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