Square Root of 277

√277 at a glance

Exact value

√277

Decimal (10 places)

16.6433169771

Rounded

16.6 · 16.64 · 16.643

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.643317

Prime factorization

277

Cube root

6.518684

How to simplify √277

277 is a prime number, so its only factors are 1 and 277. There is no perfect-square factor to pull out, which means √277 is already in its simplest radical form.

The square root of any prime is irrational. If √277 were a fraction a/b in lowest terms, then a² = 277b², so 277 would divide a — and then 277 would divide b too, contradicting “lowest terms.” That is why the decimal 16.6433169771 is only a rounded value.

Where √277 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √277 lies between 16 and 17. 277 is 21 above 256 and 12 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.6364 (0.04% low)
• Tangent from 16, i.e. 16 + 21 ÷ 32: 16.6563 (0.08% high)
• Tangent from 17, i.e. 17 − 12 ÷ 34: 16.6471 (0.02% high)

For √277 the tangent at 17 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 277 is just 12 below 289.

Finding √277 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 277: following the tangent line down to zero simplifies to averaging x with 277 ÷ x.

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 √277 = 16.6433169771 to every decimal shown.

√277 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √277 the pattern is [16; 1, 1, 1, 4, 10, 1, 7, 2, 2, 3, 3, 2, …] with the block of 21 terms after the semicolon repeating forever (only the first 12 of the 21 are shown). A pattern that never ends is one more proof that √277 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 − 277y² = 1. Its smallest solution in positive whole numbers is x = 159,150,073,798,980,475,849, y = 9,562,401,173,878,027,020 — 21 digits for x, even though 277 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 8,920,484,118² − 277 × 535,979,945² = −1.

√277 in geometry and everyday measurements

• A square patio or deck of 277 square feet is about 16.64 ft (16 ft 8 in) on each side, so edging all the way around takes 4 × √277 ≈ 66.6 ft.
• 277 = 9² + 14², so by the Pythagorean theorem √277 is the diagonal of a 9 × 14 rectangle — and the distance between the points (0, 0) and (9, 14) on a grid.

Square roots near √277 and related roots

• The cube root of 277 is about 6.518684.
• Squaring undoes the root: (√277)² = 277, while 277² = 76,729 — the number whose square root is 277.

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