Square Root of 275

√275 at a glance

Exact value

5√11

Decimal (10 places)

16.5831239518

Rounded

16.6 · 16.58 · 16.583

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.583124

Prime factorization

5² × 11

Cube root

6.502957

How to simplify √275

Look for the largest perfect square that divides 275. Here it is 25 (5²), because 275 = 25 × 11 and 11 has no square factor left:

The prime factorization tells the same story: 275 = 5² × 11. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 11 stays inside.

Check: (5√11)² = 5² × 11 = 25 × 11 = 275. As a decimal, 5√11 = 5 × 3.3166247904 ≈ 16.5831239518.

Where √275 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √275 lies between 16 and 17. 275 is 19 above 256 and 14 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.5758 (0.04% low)
• Tangent from 16, i.e. 16 + 19 ÷ 32: 16.5938 (0.06% high)
• Tangent from 17, i.e. 17 − 14 ÷ 34: 16.5882 (0.03% high)

For √275 the tangent at 17 wins, missing by only 0.0051. Tangent estimates shine when the number sits close to a perfect square — here 275 is just 14 below 289.

Finding √275 with the Babylonian method

If a guess is too big, 275 ÷ 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√275) in one step.

Start from the nearest whole number, 17 (17² = 289):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √275 = 16.5831239518 to every decimal shown.

√275 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √275 the pattern is [16; 1, 1, 2, 1, 1, 32] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √275 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 − 275y² = 1. Its smallest solution in positive whole numbers is x = 199, y = 12.

√275 in geometry and everyday measurements

• A square patio or deck of 275 square feet is about 16.58 ft (16 ft 7 in) on each side, so edging all the way around takes 4 × √275 ≈ 66.3 ft.
• 275 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √275 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 15 box, because 1² + 7² + 15² = 275.
• Since √275 = 5√11, a length of √275 is exactly 5 copies of the length √11 laid end to end.

Square roots near √275 and related roots

• The cube root of 275 is about 6.502957.
• Squaring undoes the root: (√275)² = 275, while 275² = 75,625 — the number whose square root is 275.

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