Square Root of 408

√408 at a glance

Exact value

2√102

Decimal (10 places)

20.1990098767

Rounded

20.2 · 20.20 · 20.199

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.199010

Prime factorization

2³ × 3 × 17

Cube root

7.416860

How to simplify √408

Look for the largest perfect square that divides 408. Here it is 4 (2²), because 408 = 4 × 102 and 102 has no square factor left:

The prime factorization tells the same story: 408 = 2³ × 3 × 17. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 × 17 stays inside.

Check: (2√102)² = 2² × 102 = 4 × 102 = 408. As a decimal, 2√102 = 2 × 10.0995049384 ≈ 20.1990098767.

Where √408 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √408 lies between 20 and 21. 408 is 8 above 400 and 33 below 441, so the root is closer to 20.

• Straight line between 400 and 441: 20.1951 (0.02% low)
• Tangent from 20, i.e. 20 + 8 ÷ 40: 20.2000 (0% high)
• Tangent from 21, i.e. 21 − 33 ÷ 42: 20.2143 (0.08% high)

For √408 the tangent at 20 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 408 is just 8 above 400.

Finding √408 with the Babylonian method

Picture a rectangle with an area of 408 and one side x; the other side must be 408 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √408.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √408 = 20.1990098767 to every decimal shown.

√408 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √408 the pattern is [20; 5, 40] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √408 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 − 408y² = 1. Its smallest solution in positive whole numbers is x = 101, y = 5.

√408 in geometry and everyday measurements

• A square garage floor of 408 square feet measures about 20.2 ft (20 ft 2 in) per side, and its corner-to-corner diagonal is √816 ≈ 28.6 ft.
• 408 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √408 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 20 box, because 2² + 2² + 20² = 408.
• Since √408 = 2√102, a length of √408 is exactly 2 copies of the length √102 laid end to end.

Square roots near √408 and related roots

• The cube root of 408 is about 7.416860.
• Because 408 = 4 × 102, the root is twice √102: 2 × 10.099505 ≈ 20.19901.

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