Square Root of 692

√692 at a glance

Exact value

2√173

Decimal (10 places)

26.3058928759

Rounded

26.3 · 26.31 · 26.306

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.305893

Prime factorization

2² × 173

Cube root

8.845085

How to simplify √692

Look for the largest perfect square that divides 692. Here it is 4 (2²), because 692 = 4 × 173 and 173 has no square factor left:

The prime factorization tells the same story: 692 = 2² × 173. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 173 stays inside.

Check: (2√173)² = 2² × 173 = 4 × 173 = 692. As a decimal, 2√173 = 2 × 13.152946438 ≈ 26.3058928759.

Where √692 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √692 lies between 26 and 27. 692 is 16 above 676 and 37 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.3019 (0.02% low)
• Tangent from 26, i.e. 26 + 16 ÷ 52: 26.3077 (0.01% high)
• Tangent from 27, i.e. 27 − 37 ÷ 54: 26.3148 (0.03% high)

For √692 the tangent at 26 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 692 is just 16 above 676.

Finding √692 with the Babylonian method

Picture a rectangle with an area of 692 and one side x; the other side must be 692 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √692.

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 √692 = 26.3058928759 to every decimal shown.

√692 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √692 the pattern is [26; 3, 3, 1, 2, 1, 1, 12, 1, 1, 2, 1, 3, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √692 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 − 692y² = 1. Its smallest solution in positive whole numbers is x = 2,499,849, y = 95,030.

√692 in geometry and everyday measurements

• 692 square feet is 64.3 m². Laid out as a square — a small house footprint or a lot — it is about 26.31 ft (26 ft 4 in) on a side.
• 692 = 4² + 26², so by the Pythagorean theorem √692 is the diagonal of a 4 × 26 rectangle — and the distance between the points (0, 0) and (4, 26) on a grid.
• Since √692 = 2√173, a length of √692 is exactly 2 copies of the length √173 laid end to end.

Square roots near √692 and related roots

• The cube root of 692 is about 8.845085.
• Because 692 = 4 × 173, the root is twice √173: 2 × 13.152946 ≈ 26.305893.

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