Square Root of 693

√693 at a glance

Exact value

3√77

Decimal (10 places)

26.3248931622

Rounded

26.3 · 26.32 · 26.325

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.324893

Prime factorization

3² × 7 × 11

Cube root

8.849344

How to simplify √693

Look for the largest perfect square that divides 693. Here it is 9 (3²), because 693 = 9 × 77 and 77 has no square factor left:

The prime factorization tells the same story: 693 = 3² × 7 × 11. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 7 × 11 stays inside.

Check: (3√77)² = 3² × 77 = 9 × 77 = 693. As a decimal, 3√77 = 3 × 8.7749643874 ≈ 26.3248931622.

Where √693 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √693 lies between 26 and 27. 693 is 17 above 676 and 36 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.3208 (0.02% low)
• Tangent from 26, i.e. 26 + 17 ÷ 52: 26.3269 (0.01% high)
• Tangent from 27, i.e. 27 − 36 ÷ 54: 26.3333 (0.03% high)

For √693 the tangent at 26 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 693 is just 17 above 676.

Finding √693 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 693: following the tangent line down to zero simplifies to averaging x with 693 ÷ x.

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 √693 = 26.3248931622 to every decimal shown.

√693 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √693 the pattern is [26; 3, 12, 1, 4, 1, 12, 3, 52] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √693 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 − 693y² = 1. Its smallest solution in positive whole numbers is x = 246,401, y = 9,360.

√693 in geometry and everyday measurements

• 693 square feet is 64.4 m². Laid out as a square — a small house footprint or a lot — it is about 26.32 ft (26 ft 4 in) on a side.
• 693 is not a sum of two whole-number squares — the prime factor 7 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √693 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 26 box, because 1² + 4² + 26² = 693.
• Since √693 = 3√77, a length of √693 is exactly 3 copies of the length √77 laid end to end.

Square roots near √693 and related roots

• The cube root of 693 is about 8.849344.
• Squaring undoes the root: (√693)² = 693, while 693² = 480,249 — the number whose square root is 693.

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