Square Root of 240

√240 at a glance

Exact value

4√15

Decimal (10 places)

15.4919333848

Rounded

15.5 · 15.49 · 15.492

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.491933

Prime factorization

2⁴ × 3 × 5

Cube root

6.214465

How to simplify √240

Look for the largest perfect square that divides 240. Here it is 16 (4²), because 240 = 16 × 15 and 15 has no square factor left:

The prime factorization tells the same story: 240 = 2⁴ × 3 × 5. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 3 × 5 stays inside.

240 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √240 = 2√60, and √60 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√15)² = 4² × 15 = 16 × 15 = 240. As a decimal, 4√15 = 4 × 3.8729833462 ≈ 15.4919333848.

Where √240 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √240 lies between 15 and 16. 240 is 15 above 225 and 16 below 256, so the root is closer to 15.

• Straight line between 225 and 256: 15.4839 (0.05% low)
• Tangent from 15, i.e. 15 + 15 ÷ 30: 15.5000 (0.05% high)
• Tangent from 16, i.e. 16 − 16 ÷ 32: 15.5000 (0.05% high)

For √240 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 √240 with the Babylonian method

Picture a rectangle with an area of 240 and one side x; the other side must be 240 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √240.

Start from the nearest whole number, 15 (15² = 225):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √240 = 15.4919333848 to every decimal shown.

√240 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √240 the pattern is [15; 2, 30] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √240 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 − 240y² = 1. Its smallest solution in positive whole numbers is x = 31, y = 2.

√240 in geometry and everyday measurements

• A square patio or deck of 240 square feet is about 15.49 ft (15 ft 6 in) on each side, so edging all the way around takes 4 × √240 ≈ 62 ft.
• 240 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √240 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √240 as its space diagonal.
• Since √240 = 4√15, a length of √240 is exactly 4 copies of the length √15 laid end to end.

Square roots near √240 and related roots

• The cube root of 240 is about 6.214465.
• Four times the radicand doubles the root: √960 = 2 × √240 ≈ 30.983867.

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