Square Root of 225

√225 at a glance

Exact value

15

Decimal (10 places)

15

Rounded

15

Perfect square?

Yes — 15²

Rational?

Rational (whole number)

Both square roots

±15

Prime factorization

3² × 5²

Cube root

6.082202

How to simplify √225

Write 225 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:

225 has 5 factor pairs (1 × 225, 3 × 75, 5 × 45, 9 × 25, 15 × 15). The pair with two equal numbers, 15 × 15, is what makes it a perfect square — and why it has an odd number of factors.

Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 225 = 1 + 3 + 5 + 7 + … + 29, the first 15 odd numbers.

Where √225 sits between perfect squares

225 is 15². The perfect squares on either side are 196 (14²) and 256 (16²), so the gaps are 29 below and 31 above — consecutive squares always differ by consecutive odd numbers.

Finding √225 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 225: following the tangent line down to zero simplifies to averaging x with 225 ÷ x.

To show the method at work on a perfect square, start one above the answer, at 16:

The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √225 = 15.0000000000 to every decimal shown.

√225 in geometry and everyday measurements

• A square patio or deck of 225 square feet is exactly 15 ft (15 ft) on each side, so edging all the way around takes 4 × √225 ≈ 60 ft.
• 225 = 9² + 12², so 9, 12 and 15 form a Pythagorean triple: a right triangle with legs 9 and 12 has a hypotenuse of exactly 15.

Square roots near √225 and related roots

• The cube root of 225 is about 6.082202.
• Taking the square root twice gives the fourth root: √15 ≈ 3.872983.
• Four times the radicand doubles the root: √900 = 2 × √225 ≈ 30.

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