Square Root of 916

√916 at a glance

Exact value

2√229

Decimal (10 places)

30.2654919008

Rounded

30.3 · 30.27 · 30.265

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.265492

Prime factorization

2² × 229

Cube root

9.711772

How to simplify √916

Look for the largest perfect square that divides 916. Here it is 4 (2²), because 916 = 4 × 229 and 229 has no square factor left:

The prime factorization tells the same story: 916 = 2² × 229. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 229 stays inside.

Check: (2√229)² = 2² × 229 = 4 × 229 = 916. As a decimal, 2√229 = 2 × 15.1327459504 ≈ 30.2654919008.

Where √916 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √916 lies between 30 and 31. 916 is 16 above 900 and 45 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.2623 (0.01% low)
• Tangent from 30, i.e. 30 + 16 ÷ 60: 30.2667 (0% high)
• Tangent from 31, i.e. 31 − 45 ÷ 62: 30.2742 (0.03% high)

For √916 the tangent at 30 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 916 is just 16 above 900.

Finding √916 with the Babylonian method

Picture a rectangle with an area of 916 and one side x; the other side must be 916 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √916.

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 √916 = 30.2654919008 to every decimal shown.

√916 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √916 the pattern is [30; 3, 1, 3, 3, 1, 1, 14, 1, 1, 3, 3, 1, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √916 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 − 916y² = 1. Its smallest solution in positive whole numbers is x = 5,848,201, y = 193,230.

√916 in geometry and everyday measurements

• 916 square feet is 85.1 m². Laid out as a square — a small house footprint or a lot — it is about 30.27 ft (30 ft 3 in) on a side.
• 916 = 4² + 30², so by the Pythagorean theorem √916 is the diagonal of a 4 × 30 rectangle — and the distance between the points (0, 0) and (4, 30) on a grid.
• Since √916 = 2√229, a length of √916 is exactly 2 copies of the length √229 laid end to end.

Square roots near √916 and related roots

• The cube root of 916 is about 9.711772.
• Because 916 = 4 × 229, the root is twice √229: 2 × 15.132746 ≈ 30.265492.

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