Square Root of 915

√915 at a glance

Exact value

√915

Decimal (10 places)

30.2489669245

Rounded

30.2 · 30.25 · 30.249

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.248967

Prime factorization

3 × 5 × 61

Cube root

9.708237

How to simplify √915

The prime factorization of 915 is 3 × 5 × 61. Every prime appears only once, so there is no pair to bring outside the radical — √915 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 915, 3, 5 and 61 appear an odd number of times, so √915 is irrational and 30.2489669245 is a rounded value.

Where √915 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √915 lies between 30 and 31. 915 is 15 above 900 and 46 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.2459 (0.01% low)
• Tangent from 30, i.e. 30 + 15 ÷ 60: 30.2500 (0% high)
• Tangent from 31, i.e. 31 − 46 ÷ 62: 30.2581 (0.03% high)

For √915 the tangent at 30 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 915 is just 15 above 900.

Finding √915 with the Babylonian method

If a guess is too big, 915 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√915) in one step.

Start from the nearest whole number, 30 (30² = 900):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √915 = 30.2489669245 to every decimal shown.

√915 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √915 the pattern is [30; 4, 60] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √915 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 − 915y² = 1. Its smallest solution in positive whole numbers is x = 121, y = 4.

√915 in geometry and everyday measurements

• 915 square feet is 85 m². Laid out as a square — a small house footprint or a lot — it is about 30.25 ft (30 ft 3 in) on a side.
• 915 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √915 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 17 × 25 box, because 1² + 17² + 25² = 915.

Square roots near √915 and related roots

• The cube root of 915 is about 9.708237.
• Squaring undoes the root: (√915)² = 915, while 915² = 837,225 — the number whose square root is 915.

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