Square Root of 156

√156 at a glance

Exact value

2√39

Decimal (10 places)

12.4899959968

Rounded

12.5 · 12.49 · 12.490

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.489996

Prime factorization

2² × 3 × 13

Cube root

5.383213

How to simplify √156

Look for the largest perfect square that divides 156. Here it is 4 (2²), because 156 = 4 × 39 and 39 has no square factor left:

The prime factorization tells the same story: 156 = 2² × 3 × 13. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 13 stays inside.

Check: (2√39)² = 2² × 39 = 4 × 39 = 156. As a decimal, 2√39 = 2 × 6.2449979984 ≈ 12.4899959968.

Where √156 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √156 lies between 12 and 13. 156 is 12 above 144 and 13 below 169, so the root is closer to 12.

• Straight line between 144 and 169: 12.4800 (0.08% low)
• Tangent from 12, i.e. 12 + 12 ÷ 24: 12.5000 (0.08% high)
• Tangent from 13, i.e. 13 − 13 ÷ 26: 12.5000 (0.08% high)

For √156 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 √156 with the Babylonian method

Picture a rectangle with an area of 156 and one side x; the other side must be 156 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √156.

Start from the nearest whole number, 12 (12² = 144):

The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √156 = 12.4899959968 to every decimal shown.

√156 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √156 the pattern is [12; 2, 24] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √156 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 − 156y² = 1. Its smallest solution in positive whole numbers is x = 25, y = 2.

√156 in geometry and everyday measurements

• A square room or garden bed covering 156 square feet measures about 12.49 ft (12 ft 6 in) along each wall.
• 156 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 √156 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 √156 as its space diagonal.
• Since √156 = 2√39, a length of √156 is exactly 2 copies of the length √39 laid end to end.

Square roots near √156 and related roots

• The cube root of 156 is about 5.383213.
• Four times the radicand doubles the root: √624 = 2 × √156 ≈ 24.979992.

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