Square Root of 250

√250 at a glance

Exact value

5√10

Decimal (10 places)

15.8113883008

Rounded

15.8 · 15.81 · 15.811

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.811388

Prime factorization

2 × 5³

Cube root

6.299605

How to simplify √250

Look for the largest perfect square that divides 250. Here it is 25 (5²), because 250 = 25 × 10 and 10 has no square factor left:

The prime factorization tells the same story: 250 = 2 × 5³. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 2 × 5 stays inside.

Check: (5√10)² = 5² × 10 = 25 × 10 = 250. As a decimal, 5√10 = 5 × 3.1622776602 ≈ 15.8113883008.

Where √250 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √250 lies between 15 and 16. 250 is 25 above 225 and 6 below 256, so the root is closer to 16.

• Straight line between 225 and 256: 15.8065 (0.03% low)
• Tangent from 15, i.e. 15 + 25 ÷ 30: 15.8333 (0.14% high)
• Tangent from 16, i.e. 16 − 6 ÷ 32: 15.8125 (0.01% high)

For √250 the tangent at 16 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 250 is just 6 below 256.

Finding √250 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, 16 (16² = 256):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √250 = 15.8113883008 to every decimal shown.

√250 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √250 the pattern is [15; 1, 4, 3, 3, 4, 1, 30] with the block of 7 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √250 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 − 250y² = 1. Its smallest solution in positive whole numbers is x = 39,480,499, y = 2,496,966. Because the period is odd, the equation with −1 on the right also has a solution: 4,443² − 250 × 281² = −1.

√250 in geometry and everyday measurements

• A square patio or deck of 250 square feet is about 15.81 ft (15 ft 10 in) on each side, so edging all the way around takes 4 × √250 ≈ 63.2 ft.
• 250 = 5² + 15² = 9² + 13², so by the Pythagorean theorem √250 is the diagonal of rectangles measuring 5 × 15 and 9 × 13 — and the distance between the points (0, 0) and (5, 15) on a grid.
• Since √250 = 5√10, a length of √250 is exactly 5 copies of the length √10 laid end to end.

Square roots near √250 and related roots

• The cube root of 250 is about 6.299605.
• Four times the radicand doubles the root: √1000 = 2 × √250 ≈ 31.622777.

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