Square Root of 90

√90 at a glance

Exact value

3√10

Decimal (10 places)

9.4868329805

Rounded

9.5 · 9.49 · 9.487

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.486833

Prime factorization

2 × 3² × 5

Cube root

4.481405

How to simplify √90

Look for the largest perfect square that divides 90. Here it is 9 (3²), because 90 = 9 × 10 and 10 has no square factor left:

The prime factorization tells the same story: 90 = 2 × 3² × 5. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 5 stays inside.

Check: (3√10)² = 3² × 10 = 9 × 10 = 90. As a decimal, 3√10 = 3 × 3.1622776602 ≈ 9.4868329805.

Where √90 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √90 lies between 9 and 10. 90 is 9 above 81 and 10 below 100, so the root is closer to 9.

• Straight line between 81 and 100: 9.4737 (0.14% low)
• Tangent from 9, i.e. 9 + 9 ÷ 18: 9.5000 (0.14% high)
• Tangent from 10, i.e. 10 − 10 ÷ 20: 9.5000 (0.14% high)

For √90 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 √90 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, 9 (9² = 81):

The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √90 = 9.4868329805 to every decimal shown.

√90 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √90 the pattern is [9; 2, 18] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √90 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 − 90y² = 1. Its smallest solution in positive whole numbers is x = 19, y = 2.

√90 in geometry and everyday measurements

• A square room or garden bed covering 90 square feet measures about 9.49 ft (9 ft 6 in) along each wall.
• 90 = 3² + 9², so by the Pythagorean theorem √90 is the diagonal of a 3 × 9 rectangle — and the distance between the points (0, 0) and (3, 9) on a grid.
• Since √90 = 3√10, a length of √90 is exactly 3 copies of the length √10 laid end to end.

Square roots near √90 and related roots

• The cube root of 90 is about 4.481405.
• Four times the radicand doubles the root: √360 = 2 × √90 ≈ 18.973666.

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