Square Root of 236

√236 at a glance

Exact value

2√59

Decimal (10 places)

15.3622914957

Rounded

15.4 · 15.36 · 15.362

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.362291

Prime factorization

2² × 59

Cube root

6.179747

How to simplify √236

Look for the largest perfect square that divides 236. Here it is 4 (2²), because 236 = 4 × 59 and 59 has no square factor left:

The prime factorization tells the same story: 236 = 2² × 59. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 59 stays inside.

Check: (2√59)² = 2² × 59 = 4 × 59 = 236. As a decimal, 2√59 = 2 × 7.6811457479 ≈ 15.3622914957.

Where √236 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √236 lies between 15 and 16. 236 is 11 above 225 and 20 below 256, so the root is closer to 15.

• Straight line between 225 and 256: 15.3548 (0.05% low)
• Tangent from 15, i.e. 15 + 11 ÷ 30: 15.3667 (0.03% high)
• Tangent from 16, i.e. 16 − 20 ÷ 32: 15.3750 (0.08% high)

For √236 the tangent at 15 wins, missing by only 0.0044. Tangent estimates shine when the number sits close to a perfect square — here 236 is just 11 above 225.

Finding √236 with the Babylonian method

Picture a rectangle with an area of 236 and one side x; the other side must be 236 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √236.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √236 = 15.3622914957 to every decimal shown.

√236 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √236 the pattern is [15; 2, 1, 3, 5, 1, 6, 1, 5, 3, 1, 2, 30] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √236 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 − 236y² = 1. Its smallest solution in positive whole numbers is x = 561,799, y = 36,570.

√236 in geometry and everyday measurements

• A square patio or deck of 236 square feet is about 15.36 ft (15 ft 4 in) on each side, so edging all the way around takes 4 × √236 ≈ 61.4 ft.
• 236 is not a sum of two whole-number squares — the prime factor 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √236 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 14 box, because 2² + 6² + 14² = 236.
• Since √236 = 2√59, a length of √236 is exactly 2 copies of the length √59 laid end to end.

Square roots near √236 and related roots

• The cube root of 236 is about 6.179747.
• Four times the radicand doubles the root: √944 = 2 × √236 ≈ 30.724583.

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