Square Root of 60

√60 at a glance

Exact value

2√15

Decimal (10 places)

7.7459666924

Rounded

7.7 · 7.75 · 7.746

Perfect square?

No — between 7² and 8²

Rational?

Irrational

Both square roots

±7.745967

Prime factorization

2² × 3 × 5

Cube root

3.914868

How to simplify √60

Look for the largest perfect square that divides 60. Here it is 4 (2²), because 60 = 4 × 15 and 15 has no square factor left:

The prime factorization tells the same story: 60 = 2² × 3 × 5. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 5 stays inside.

Check: (2√15)² = 2² × 15 = 4 × 15 = 60. As a decimal, 2√15 = 2 × 3.8729833462 ≈ 7.7459666924.

Where √60 sits between perfect squares

49 = 7² and 64 = 8² are the nearest perfect squares, so √60 lies between 7 and 8. 60 is 11 above 49 and 4 below 64, so the root is closer to 8.

• Straight line between 49 and 64: 7.7333 (0.16% low)
• Tangent from 7, i.e. 7 + 11 ÷ 14: 7.7857 (0.51% high)
• Tangent from 8, i.e. 8 − 4 ÷ 16: 7.7500 (0.05% high)

For √60 the tangent at 8 wins, missing by only 0.004. Tangent estimates shine when the number sits close to a perfect square — here 60 is just 4 below 64.

Finding √60 with the Babylonian method

Picture a rectangle with an area of 60 and one side x; the other side must be 60 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √60.

Start from the nearest whole number, 8 (8² = 64):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √60 = 7.7459666924 to every decimal shown.

√60 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √60 the pattern is [7; 1, 2, 1, 14] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √60 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 − 60y² = 1. Its smallest solution in positive whole numbers is x = 31, y = 4.

√60 in geometry and everyday measurements

• A square room or garden bed covering 60 square feet measures about 7.75 ft (7 ft 9 in) along each wall.
• 60 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 √60 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 √60 as its space diagonal.
• Since √60 = 2√15, a length of √60 is exactly 2 copies of the length √15 laid end to end.

Square roots near √60 and related roots

• The cube root of 60 is about 3.914868.
• Four times the radicand doubles the root: √240 = 2 × √60 ≈ 15.491933.

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