Square Root of 680

√680 at a glance

Exact value

2√170

Decimal (10 places)

26.0768096208

Rounded

26.1 · 26.08 · 26.077

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.076810

Prime factorization

2³ × 5 × 17

Cube root

8.793659

How to simplify √680

Look for the largest perfect square that divides 680. Here it is 4 (2²), because 680 = 4 × 170 and 170 has no square factor left:

The prime factorization tells the same story: 680 = 2³ × 5 × 17. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 5 × 17 stays inside.

Check: (2√170)² = 2² × 170 = 4 × 170 = 680. As a decimal, 2√170 = 2 × 13.0384048104 ≈ 26.0768096208.

Where √680 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √680 lies between 26 and 27. 680 is 4 above 676 and 49 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.0755 (0.01% low)
• Tangent from 26, i.e. 26 + 4 ÷ 52: 26.0769 (0% high)
• Tangent from 27, i.e. 27 − 49 ÷ 54: 26.0926 (0.06% high)

For √680 the tangent at 26 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 680 is just 4 above 676.

Finding √680 with the Babylonian method

Picture a rectangle with an area of 680 and one side x; the other side must be 680 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √680.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √680 = 26.0768096208 to every decimal shown.

√680 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √680 the pattern is [26; 13, 52] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √680 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 − 680y² = 1. Its smallest solution in positive whole numbers is x = 339, y = 13.

√680 in geometry and everyday measurements

• 680 square feet is 63.2 m². Laid out as a square — a small house footprint or a lot — it is about 26.08 ft (26 ft 1 in) on a side.
• 680 = 2² + 26² = 14² + 22², so by the Pythagorean theorem √680 is the diagonal of rectangles measuring 2 × 26 and 14 × 22 — and the distance between the points (0, 0) and (2, 26) on a grid.
• Since √680 = 2√170, a length of √680 is exactly 2 copies of the length √170 laid end to end.

Square roots near √680 and related roots

• The cube root of 680 is about 8.793659.
• Because 680 = 4 × 170, the root is twice √170: 2 × 13.038405 ≈ 26.07681.

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