Square Root of 315

√315 at a glance

Exact value

3√35

Decimal (10 places)

17.7482393493

Rounded

17.7 · 17.75 · 17.748

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.748239

Prime factorization

3² × 5 × 7

Cube root

6.804092

How to simplify √315

Look for the largest perfect square that divides 315. Here it is 9 (3²), because 315 = 9 × 35 and 35 has no square factor left:

The prime factorization tells the same story: 315 = 3² × 5 × 7. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 5 × 7 stays inside.

Check: (3√35)² = 3² × 35 = 9 × 35 = 315. As a decimal, 3√35 = 3 × 5.9160797831 ≈ 17.7482393493.

Where √315 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √315 lies between 17 and 18. 315 is 26 above 289 and 9 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.7429 (0.03% low)
• Tangent from 17, i.e. 17 + 26 ÷ 34: 17.7647 (0.09% high)
• Tangent from 18, i.e. 18 − 9 ÷ 36: 17.7500 (0.01% high)

For √315 the tangent at 18 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 315 is just 9 below 324.

Finding √315 with the Babylonian method

If a guess is too big, 315 ÷ 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√315) in one step.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √315 = 17.7482393493 to every decimal shown.

√315 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √315 the pattern is [17; 1, 2, 1, 34] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √315 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 − 315y² = 1. Its smallest solution in positive whole numbers is x = 71, y = 4.

√315 in geometry and everyday measurements

• A square patio or deck of 315 square feet is about 17.75 ft (17 ft 9 in) on each side, so edging all the way around takes 4 × √315 ≈ 71 ft.
• 315 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √315 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 17 box, because 1² + 5² + 17² = 315.
• Since √315 = 3√35, a length of √315 is exactly 3 copies of the length √35 laid end to end.

Square roots near √315 and related roots

• The cube root of 315 is about 6.804092.
• Squaring undoes the root: (√315)² = 315, while 315² = 99,225 — the number whose square root is 315.

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