Square Root of 684

√684 at a glance

Exact value

6√19

Decimal (10 places)

26.1533936612

Rounded

26.2 · 26.15 · 26.153

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.153394

Prime factorization

2² × 3² × 19

Cube root

8.810868

How to simplify √684

Look for the largest perfect square that divides 684. Here it is 36 (6²), because 684 = 36 × 19 and 19 has no square factor left:

The prime factorization tells the same story: 684 = 2² × 3² × 19. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 19 stays inside.

684 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √684 = 2√171, and √171 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√19)² = 6² × 19 = 36 × 19 = 684. As a decimal, 6√19 = 6 × 4.3588989435 ≈ 26.1533936612.

Where √684 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √684 lies between 26 and 27. 684 is 8 above 676 and 45 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.1509 (0.01% low)
• Tangent from 26, i.e. 26 + 8 ÷ 52: 26.1538 (0% high)
• Tangent from 27, i.e. 27 − 45 ÷ 54: 26.1667 (0.05% high)

For √684 the tangent at 26 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 684 is just 8 above 676.

Finding √684 with the Babylonian method

Picture a rectangle with an area of 684 and one side x; the other side must be 684 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √684.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √684 = 26.1533936612 to every decimal shown.

√684 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √684 the pattern is [26; 6, 1, 1, 12, 1, 1, 6, 52] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √684 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 − 684y² = 1. Its smallest solution in positive whole numbers is x = 57,799, y = 2,210.

√684 in geometry and everyday measurements

• 684 square feet is 63.5 m². Laid out as a square — a small house footprint or a lot — it is about 26.15 ft (26 ft 2 in) on a side.
• 684 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √684 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 26 box, because 2² + 2² + 26² = 684.
• Since √684 = 6√19, a length of √684 is exactly 6 copies of the length √19 laid end to end.

Square roots near √684 and related roots

• The cube root of 684 is about 8.810868.
• Because 684 = 4 × 171, the root is twice √171: 2 × 13.076697 ≈ 26.153394.

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