Square Root of 245

√245 at a glance

Exact value

7√5

Decimal (10 places)

15.6524758425

Rounded

15.7 · 15.65 · 15.652

Perfect square?

No — between 15² and 16²

Rational?

Irrational

Both square roots

±15.652476

Prime factorization

5 × 7²

Cube root

6.257325

How to simplify √245

Look for the largest perfect square that divides 245. Here it is 49 (7²), because 245 = 49 × 5 and 5 has no square factor left:

The prime factorization tells the same story: 245 = 5 × 7². Each pair of equal primes leaves the radical as one factor, so 7 comes out and 5 stays inside.

Check: (7√5)² = 7² × 5 = 49 × 5 = 245. As a decimal, 7√5 = 7 × 2.2360679775 ≈ 15.6524758425.

Where √245 sits between perfect squares

225 = 15² and 256 = 16² are the nearest perfect squares, so √245 lies between 15 and 16. 245 is 20 above 225 and 11 below 256, so the root is closer to 16.

• Straight line between 225 and 256: 15.6452 (0.05% low)
• Tangent from 15, i.e. 15 + 20 ÷ 30: 15.6667 (0.09% high)
• Tangent from 16, i.e. 16 − 11 ÷ 32: 15.6563 (0.02% high)

For √245 the tangent at 16 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 245 is just 11 below 256.

Finding √245 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 245: following the tangent line down to zero simplifies to averaging x with 245 ÷ x.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √245 = 15.6524758425 to every decimal shown.

√245 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √245 the pattern is [15; 1, 1, 1, 7, 6, 7, 1, 1, 1, 30] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √245 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 − 245y² = 1. Its smallest solution in positive whole numbers is x = 51,841, y = 3,312.

√245 in geometry and everyday measurements

• A square patio or deck of 245 square feet is about 15.65 ft (15 ft 8 in) on each side, so edging all the way around takes 4 × √245 ≈ 62.6 ft.
• 245 = 7² + 14², so by the Pythagorean theorem √245 is the diagonal of a 7 × 14 rectangle — and the distance between the points (0, 0) and (7, 14) on a grid.
• Since √245 = 7√5, a length of √245 is exactly 7 copies of the length √5 laid end to end.

Square roots near √245 and related roots

• The cube root of 245 is about 6.257325.
• Four times the radicand doubles the root: √980 = 2 × √245 ≈ 31.304952.

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