Square Root of 10

√10 at a glance

Exact value

√10

Decimal (10 places)

3.1622776602

Rounded

3.2 · 3.16 · 3.162

Perfect square?

No — between 3² and 4²

Rational?

Irrational

Both square roots

±3.162278

Prime factorization

2 × 5

Cube root

2.154435

How to simplify √10

The prime factorization of 10 is 2 × 5. Every prime appears only once, so there is no pair to bring outside the radical — √10 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 10, 2 and 5 appear an odd number of times, so √10 is irrational and 3.1622776602 is a rounded value.

Where √10 sits between perfect squares

9 = 3² and 16 = 4² are the nearest perfect squares, so √10 lies between 3 and 4. 10 is 1 above 9 and 6 below 16, so the root is closer to 3.

• Straight line between 9 and 16: 3.1429 (0.61% low)
• Tangent from 3, i.e. 3 + 1 ÷ 6: 3.1667 (0.14% high)
• Tangent from 4, i.e. 4 − 6 ÷ 8: 3.2500 (2.77% high)

For √10 the tangent at 3 wins, missing by only 0.0044. Tangent estimates shine when the number sits close to a perfect square — here 10 is just 1 above 9.

Finding √10 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, 3 (3² = 9):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √10 = 3.1622776602 to every decimal shown.

√10 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √10 the pattern is [3; 6] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 10 is one more than a perfect square (3² + 1). A pattern that never ends is one more proof that √10 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 − 10y² = 1. Its smallest solution in positive whole numbers is x = 19, y = 6. Because the period is odd, the equation with −1 on the right also has a solution: 3² − 10 × 1² = −1.

√10 in geometry and everyday measurements

• A square tile with an area of 10 square inches has sides about 3.162 in long.
• 10 = 1² + 3², so by the Pythagorean theorem √10 is the diagonal of a 1 × 3 rectangle — and the distance between the points (0, 0) and (1, 3) on a grid.

√10 ≈ 3.1623 is the halfway point on a logarithmic scale between 1 and 10 (10 to the power ½), so it marks the middle of each decade on log-scale graph paper and slide rules.

Square roots near √10 and related roots

• The cube root of 10 is about 2.154435.
• Multiplying the radicand by 100 multiplies the root by 10: √1,000 = 10 × √10 ≈ 31.622777.

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