Square Root of 172

√172 at a glance

Exact value

2√43

Decimal (10 places)

13.1148770486

Rounded

13.1 · 13.11 · 13.115

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.114877

Prime factorization

2² × 43

Cube root

5.561298

How to simplify √172

Look for the largest perfect square that divides 172. Here it is 4 (2²), because 172 = 4 × 43 and 43 has no square factor left:

The prime factorization tells the same story: 172 = 2² × 43. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 43 stays inside.

Check: (2√43)² = 2² × 43 = 4 × 43 = 172. As a decimal, 2√43 = 2 × 6.5574385243 ≈ 13.1148770486.

Where √172 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √172 lies between 13 and 14. 172 is 3 above 169 and 24 below 196, so the root is closer to 13.

• Straight line between 169 and 196: 13.1111 (0.03% low)
• Tangent from 13, i.e. 13 + 3 ÷ 26: 13.1154 (0% high)
• Tangent from 14, i.e. 14 − 24 ÷ 28: 13.1429 (0.21% high)

For √172 the tangent at 13 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 172 is just 3 above 169.

Finding √172 with the Babylonian method

Picture a rectangle with an area of 172 and one side x; the other side must be 172 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √172.

Start from the nearest whole number, 13 (13² = 169):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √172 = 13.1148770486 to every decimal shown.

√172 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √172 the pattern is [13; 8, 1, 2, 2, 1, 1, 3, 6, 3, 1, 1, 2, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √172 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 − 172y² = 1. Its smallest solution in positive whole numbers is x = 24,248,647, y = 1,848,942.

√172 in geometry and everyday measurements

• A square room or garden bed covering 172 square feet measures about 13.11 ft (13 ft 1 in) along each wall.
• 172 is not a sum of two whole-number squares — the prime factor 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √172 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 6 × 10 box, because 6² + 6² + 10² = 172.
• Since √172 = 2√43, a length of √172 is exactly 2 copies of the length √43 laid end to end.

Square roots near √172 and related roots

• The cube root of 172 is about 5.561298.
• Four times the radicand doubles the root: √688 = 2 × √172 ≈ 26.229754.

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