Square Root of 5

√5 at a glance

Exact value

√5

Decimal (10 places)

2.2360679775

Rounded

2.2 · 2.24 · 2.236

Perfect square?

No — between 2² and 3²

Rational?

Irrational

Both square roots

±2.236068

Prime factorization

5

Cube root

1.709976

How to simplify √5

5 is a prime number, so its only factors are 1 and 5. There is no perfect-square factor to pull out, which means √5 is already in its simplest radical form.

The square root of any prime is irrational. If √5 were a fraction a/b in lowest terms, then a² = 5b², so 5 would divide a — and then 5 would divide b too, contradicting “lowest terms.” That is why the decimal 2.2360679775 is only a rounded value.

Where √5 sits between perfect squares

4 = 2² and 9 = 3² are the nearest perfect squares, so √5 lies between 2 and 3. 5 is 1 above 4 and 4 below 9, so the root is closer to 2.

• Straight line between 4 and 9: 2.2000 (1.61% low)
• Tangent from 2, i.e. 2 + 1 ÷ 4: 2.2500 (0.62% high)
• Tangent from 3, i.e. 3 − 4 ÷ 6: 2.3333 (4.35% high)

For √5 the tangent at 2 wins, missing by only 0.0139. Tangent estimates shine when the number sits close to a perfect square — here 5 is just 1 above 4.

Finding √5 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 5: following the tangent line down to zero simplifies to averaging x with 5 ÷ x.

Start from the nearest whole number, 2 (2² = 4):

The count of correct decimals went 1, 4, 9 and all 10 over 4 steps — roughly doubling each time — until the guess matched √5 = 2.2360679775 to every decimal shown.

√5 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √5 the pattern is [2; 4] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 5 is one more than a perfect square (2² + 1). A pattern that never ends is one more proof that √5 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 − 5y² = 1. Its smallest solution in positive whole numbers is x = 9, y = 4. Because the period is odd, the equation with −1 on the right also has a solution: 2² − 5 × 1² = −1.

√5 in geometry and everyday measurements

• A square tile with an area of 5 square inches has sides about 2.236 in long.
• 5 = 1² + 2², so by the Pythagorean theorem √5 is the diagonal of a 1 × 2 rectangle — and the distance between the points (0, 0) and (1, 2) on a grid.

√5 sits inside the golden ratio: φ = (1 + √5) ÷ 2 ≈ 1.6180339887. It is also the diagonal of a 1 × 2 rectangle.

Square roots near √5 and related roots

• The cube root of 5 is about 1.709976.
• Multiplying the radicand by 100 multiplies the root by 10: √500 = 10 × √5 ≈ 22.36068.

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