Square Root of 691

√691 at a glance

Exact value

√691

Decimal (10 places)

26.2868788562

Rounded

26.3 · 26.29 · 26.287

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.286879

Prime factorization

691

Cube root

8.840823

How to simplify √691

691 is a prime number, so its only factors are 1 and 691. There is no perfect-square factor to pull out, which means √691 is already in its simplest radical form.

The square root of any prime is irrational. If √691 were a fraction a/b in lowest terms, then a² = 691b², so 691 would divide a — and then 691 would divide b too, contradicting “lowest terms.” That is why the decimal 26.2868788562 is only a rounded value.

Where √691 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √691 lies between 26 and 27. 691 is 15 above 676 and 38 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.2830 (0.01% low)
• Tangent from 26, i.e. 26 + 15 ÷ 52: 26.2885 (0.01% high)
• Tangent from 27, i.e. 27 − 38 ÷ 54: 26.2963 (0.04% high)

For √691 the tangent at 26 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 691 is just 15 above 676.

Finding √691 with the Babylonian method

If a guess is too big, 691 ÷ 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√691) in one step.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √691 = 26.2868788562 to every decimal shown.

√691 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √691 the pattern is [26; 3, 2, 17, 10, 2, 5, 2, 1, 2, 1, 4, 1, …] with the block of 38 terms after the semicolon repeating forever (only the first 12 of the 38 are shown). A pattern that never ends is one more proof that √691 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 − 691y² = 1. Its smallest solution in positive whole numbers is x = 31,138,100,617,500,578,690, y = 1,184,549,173,291,009,383 — 20 digits for x, even though 691 is small, which is what makes Pell’s equation famous.

√691 in geometry and everyday measurements

• 691 square feet is 64.2 m². Laid out as a square — a small house footprint or a lot — it is about 26.29 ft (26 ft 3 in) on a side.
• 691 is not a sum of two whole-number squares — 691 is itself a prime that is one less than a multiple of 4, which rules that out — so √691 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 15 × 21 box, because 5² + 15² + 21² = 691.

Square roots near √691 and related roots

• The cube root of 691 is about 8.840823.
• Squaring undoes the root: (√691)² = 691, while 691² = 477,481 — the number whose square root is 691.

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