Square Root of 243

√243 at a glance

Exact value

9√3

Decimal (10 places)

15.5884572681

Rounded

15.6 · 15.59 · 15.588

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.588457

Prime factorization

3⁵

Cube root

6.240251

How to simplify √243

Look for the largest perfect square that divides 243. Here it is 81 (9²), because 243 = 81 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 243 = 3⁵. Each pair of equal primes leaves the radical as one factor, so 3² comes out and 3 stays inside.

243 has 2 square factors (9 and 81). Starting with a smaller one still works but takes more rounds: √243 = 3√27, and √27 can be simplified again. Using 81 straight away finishes in one step.

Check: (9√3)² = 9² × 3 = 81 × 3 = 243. As a decimal, 9√3 = 9 × 1.7320508076 ≈ 15.5884572681.

Where √243 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √243 lies between 15 and 16. 243 is 18 above 225 and 13 below 256, so the root is closer to 16.

• Straight line between 225 and 256: 15.5806 (0.05% low)
• Tangent from 15, i.e. 15 + 18 ÷ 30: 15.6000 (0.07% high)
• Tangent from 16, i.e. 16 − 13 ÷ 32: 15.5938 (0.03% high)

For √243 the tangent at 16 wins, missing by only 0.0053. Tangent estimates shine when the number sits close to a perfect square — here 243 is just 13 below 256.

Finding √243 with the Babylonian method

If a guess is too big, 243 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√243) in one step.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √243 = 15.5884572681 to every decimal shown.

√243 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √243 the pattern is [15; 1, 1, 2, 3, 15, 3, 2, 1, 1, 30] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √243 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 − 243y² = 1. Its smallest solution in positive whole numbers is x = 70,226, y = 4,505.

√243 in geometry and everyday measurements

• A square patio or deck of 243 square feet is about 15.59 ft (15 ft 7 in) on each side, so edging all the way around takes 4 × √243 ≈ 62.4 ft.
• 243 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 √243 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 11 × 11 box, because 1² + 11² + 11² = 243.
• Since √243 = 9√3, a length of √243 is exactly 9 copies of the length √3 laid end to end.

Square roots near √243 and related roots

• The cube root of 243 is about 6.240251.
• Four times the radicand doubles the root: √972 = 2 × √243 ≈ 31.176915.

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