Square Root of 420

√420 at a glance

Exact value

2√105

Decimal (10 places)

20.4939015319

Rounded

20.5 · 20.49 · 20.494

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.493902

Prime factorization

2² × 3 × 5 × 7

Cube root

7.488872

How to simplify √420

Look for the largest perfect square that divides 420. Here it is 4 (2²), because 420 = 4 × 105 and 105 has no square factor left:

The prime factorization tells the same story: 420 = 2² × 3 × 5 × 7. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 5 × 7 stays inside.

Check: (2√105)² = 2² × 105 = 4 × 105 = 420. As a decimal, 2√105 = 2 × 10.246950766 ≈ 20.4939015319.

Where √420 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √420 lies between 20 and 21. 420 is 20 above 400 and 21 below 441, so the root is closer to 20.

• Straight line between 400 and 441: 20.4878 (0.03% low)
• Tangent from 20, i.e. 20 + 20 ÷ 40: 20.5000 (0.03% high)
• Tangent from 21, i.e. 21 − 21 ÷ 42: 20.5000 (0.03% high)

For √420 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √420 with the Babylonian method

Picture a rectangle with an area of 420 and one side x; the other side must be 420 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √420.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √420 = 20.4939015319 to every decimal shown.

√420 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √420 the pattern is [20; 2, 40] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √420 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 − 420y² = 1. Its smallest solution in positive whole numbers is x = 41, y = 2.

√420 in geometry and everyday measurements

• A square garage floor of 420 square feet measures about 20.49 ft (20 ft 6 in) per side, and its corner-to-corner diagonal is √840 ≈ 29 ft.
• 420 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √420 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 20 box, because 2² + 4² + 20² = 420.
• Since √420 = 2√105, a length of √420 is exactly 2 copies of the length √105 laid end to end.

Square roots near √420 and related roots

• The cube root of 420 is about 7.488872.
• Because 420 = 4 × 105, the root is twice √105: 2 × 10.246951 ≈ 20.493902.

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