Square Root of 661

√661 at a glance

Exact value

√661

Decimal (10 places)

25.7099202644

Rounded

25.7 · 25.71 · 25.710

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.709920

Prime factorization

661

Cube root

8.710983

How to simplify √661

661 is a prime number, so its only factors are 1 and 661. There is no perfect-square factor to pull out, which means √661 is already in its simplest radical form.

The square root of any prime is irrational. If √661 were a fraction a/b in lowest terms, then a² = 661b², so 661 would divide a — and then 661 would divide b too, contradicting “lowest terms.” That is why the decimal 25.7099202644 is only a rounded value.

Where √661 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √661 lies between 25 and 26. 661 is 36 above 625 and 15 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.7059 (0.02% low)
• Tangent from 25, i.e. 25 + 36 ÷ 50: 25.7200 (0.04% high)
• Tangent from 26, i.e. 26 − 15 ÷ 52: 25.7115 (0.01% high)

For √661 the tangent at 26 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 661 is just 15 below 676.

Finding √661 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 661: following the tangent line down to zero simplifies to averaging x with 661 ÷ 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 √661 = 25.7099202644 to every decimal shown.

√661 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √661 the pattern is [25; 1, 2, 2, 4, 4, 16, 1, 9, 2, 1, 12, 5, …] with the block of 39 terms after the semicolon repeating forever (only the first 12 of the 39 are shown). A pattern that never ends is one more proof that √661 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 − 661y² = 1. Its smallest solution in positive whole numbers is x = 16,421,658,242,965,910,275,055,840,472,270,471,049, y = 638,728,478,116,949,861,246,791,167,518,480,580 — 38 digits for x, even though 661 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 2,865,454,435,422,583,218² − 661 × 111,453,260,296,346,905² = −1.

√661 in geometry and everyday measurements

• 661 square feet is 61.4 m². Laid out as a square — a small house footprint or a lot — it is about 25.71 ft (25 ft 9 in) on a side.
• 661 = 6² + 25², so by the Pythagorean theorem √661 is the diagonal of a 6 × 25 rectangle — and the distance between the points (0, 0) and (6, 25) on a grid.

Square roots near √661 and related roots

• The cube root of 661 is about 8.710983.
• Squaring undoes the root: (√661)² = 661, while 661² = 436,921 — the number whose square root is 661.

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