Square Root of 40

√40 at a glance

Exact value

2√10

Decimal (10 places)

6.3245553203

Rounded

6.3 · 6.32 · 6.325

Perfect square?

No — between 6² and 7²

Rational?

Irrational

Both square roots

±6.324555

Prime factorization

2³ × 5

Cube root

3.419952

How to simplify √40

Look for the largest perfect square that divides 40. Here it is 4 (2²), because 40 = 4 × 10 and 10 has no square factor left:

The prime factorization tells the same story: 40 = 2³ × 5. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 5 stays inside.

Check: (2√10)² = 2² × 10 = 4 × 10 = 40. As a decimal, 2√10 = 2 × 3.1622776602 ≈ 6.3245553203.

Where √40 sits between perfect squares

36 = 6² and 49 = 7² are the nearest perfect squares, so √40 lies between 6 and 7. 40 is 4 above 36 and 9 below 49, so the root is closer to 6.

• Straight line between 36 and 49: 6.3077 (0.27% low)
• Tangent from 6, i.e. 6 + 4 ÷ 12: 6.3333 (0.14% high)
• Tangent from 7, i.e. 7 − 9 ÷ 14: 6.3571 (0.52% high)

For √40 the tangent at 6 wins, missing by only 0.0088. Tangent estimates shine when the number sits close to a perfect square — here 40 is just 4 above 36.

Finding √40 with the Babylonian method

Picture a rectangle with an area of 40 and one side x; the other side must be 40 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √40.

Start from the nearest whole number, 6 (6² = 36):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √40 = 6.3245553203 to every decimal shown.

√40 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √40 the pattern is [6; 3, 12] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √40 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 − 40y² = 1. Its smallest solution in positive whole numbers is x = 19, y = 3.

√40 in geometry and everyday measurements

• A square room or garden bed covering 40 square feet measures about 6.32 ft (6 ft 4 in) along each wall.
• 40 = 2² + 6², so by the Pythagorean theorem √40 is the diagonal of a 2 × 6 rectangle — and the distance between the points (0, 0) and (2, 6) on a grid.
• Since √40 = 2√10, a length of √40 is exactly 2 copies of the length √10 laid end to end.

Square roots near √40 and related roots

• The cube root of 40 is about 3.419952.
• Four times the radicand doubles the root: √160 = 2 × √40 ≈ 12.649111.

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