Square Root of 650

√650 at a glance

Exact value

5√26

Decimal (10 places)

25.4950975680

Rounded

25.5 · 25.50 · 25.495

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.495098

Prime factorization

2 × 5² × 13

Cube root

8.662391

How to simplify √650

Look for the largest perfect square that divides 650. Here it is 25 (5²), because 650 = 25 × 26 and 26 has no square factor left:

The prime factorization tells the same story: 650 = 2 × 5² × 13. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 2 × 13 stays inside.

Check: (5√26)² = 5² × 26 = 25 × 26 = 650. As a decimal, 5√26 = 5 × 5.0990195136 ≈ 25.4950975680.

Where √650 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √650 lies between 25 and 26. 650 is 25 above 625 and 26 below 676, so the root is closer to 25.

• Straight line between 625 and 676: 25.4902 (0.02% low)
• Tangent from 25, i.e. 25 + 25 ÷ 50: 25.5000 (0.02% high)
• Tangent from 26, i.e. 26 − 26 ÷ 52: 25.5000 (0.02% high)

For √650 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √650 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 25 (25² = 625):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √650 = 25.4950975680 to every decimal shown.

√650 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √650 the pattern is [25; 2, 50] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √650 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 − 650y² = 1. Its smallest solution in positive whole numbers is x = 51, y = 2.

√650 in geometry and everyday measurements

• A square garage floor of 650 square feet measures about 25.5 ft (25 ft 6 in) per side, and its corner-to-corner diagonal is √1300 ≈ 36.1 ft.
• 650 = 5² + 25² = 11² + 23² = 17² + 19², so by the Pythagorean theorem √650 is the diagonal of rectangles measuring 5 × 25, 11 × 23 and 17 × 19 — and the distance between the points (0, 0) and (5, 25) on a grid.
• Since √650 = 5√26, a length of √650 is exactly 5 copies of the length √26 laid end to end.

Square roots near √650 and related roots

• The cube root of 650 is about 8.662391.
• Squaring undoes the root: (√650)² = 650, while 650² = 422,500 — the number whose square root is 650.

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