Square Root of 20

√20 at a glance

Exact value

2√5

Decimal (10 places)

4.4721359550

Rounded

4.5 · 4.47 · 4.472

Perfect square?

No — between 4² and 5²

Rational?

Irrational

Both square roots

±4.472136

Prime factorization

2² × 5

Cube root

2.714418

How to simplify √20

Look for the largest perfect square that divides 20. Here it is 4 (2²), because 20 = 4 × 5 and 5 has no square factor left:

The prime factorization tells the same story: 20 = 2² × 5. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 5 stays inside.

Check: (2√5)² = 2² × 5 = 4 × 5 = 20. As a decimal, 2√5 = 2 × 2.2360679775 ≈ 4.4721359550.

Where √20 sits between perfect squares

16 = 4² and 25 = 5² are the nearest perfect squares, so √20 lies between 4 and 5. 20 is 4 above 16 and 5 below 25, so the root is closer to 4.

• Straight line between 16 and 25: 4.4444 (0.62% low)
• Tangent from 4, i.e. 4 + 4 ÷ 8: 4.5000 (0.62% high)
• Tangent from 5, i.e. 5 − 5 ÷ 10: 4.5000 (0.62% high)

For √20 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 √20 with the Babylonian method

Picture a rectangle with an area of 20 and one side x; the other side must be 20 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √20.

Start from the nearest whole number, 4 (4² = 16):

The count of correct decimals went 1, 4, 9 and all 10 over 4 steps — roughly doubling each time — until the guess matched √20 = 4.4721359550 to every decimal shown.

√20 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √20 the pattern is [4; 2, 8] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √20 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 − 20y² = 1. Its smallest solution in positive whole numbers is x = 9, y = 2.

√20 in geometry and everyday measurements

• A square tile with an area of 20 square inches has sides about 4.472 in long.
• 20 = 2² + 4², so by the Pythagorean theorem √20 is the diagonal of a 2 × 4 rectangle — and the distance between the points (0, 0) and (2, 4) on a grid.
• Since √20 = 2√5, a length of √20 is exactly 2 copies of the length √5 laid end to end.

Square roots near √20 and related roots

• The cube root of 20 is about 2.714418.
• Four times the radicand doubles the root: √80 = 2 × √20 ≈ 8.944272.

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