Square Root of 276

√276 at a glance

Exact value

2√69

Decimal (10 places)

16.6132477258

Rounded

16.6 · 16.61 · 16.613

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.613248

Prime factorization

2² × 3 × 23

Cube root

6.510830

How to simplify √276

Look for the largest perfect square that divides 276. Here it is 4 (2²), because 276 = 4 × 69 and 69 has no square factor left:

The prime factorization tells the same story: 276 = 2² × 3 × 23. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 23 stays inside.

Check: (2√69)² = 2² × 69 = 4 × 69 = 276. As a decimal, 2√69 = 2 × 8.3066238629 ≈ 16.6132477258.

Where √276 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √276 lies between 16 and 17. 276 is 20 above 256 and 13 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.6061 (0.04% low)
• Tangent from 16, i.e. 16 + 20 ÷ 32: 16.6250 (0.07% high)
• Tangent from 17, i.e. 17 − 13 ÷ 34: 16.6176 (0.03% high)

For √276 the tangent at 17 wins, missing by only 0.0044. Tangent estimates shine when the number sits close to a perfect square — here 276 is just 13 below 289.

Finding √276 with the Babylonian method

Picture a rectangle with an area of 276 and one side x; the other side must be 276 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √276.

Start from the nearest whole number, 17 (17² = 289):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √276 = 16.6132477258 to every decimal shown.

√276 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √276 the pattern is [16; 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 32] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √276 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 − 276y² = 1. Its smallest solution in positive whole numbers is x = 7,775, y = 468.

√276 in geometry and everyday measurements

• A square patio or deck of 276 square feet is about 16.61 ft (16 ft 7 in) on each side, so edging all the way around takes 4 × √276 ≈ 66.5 ft.
• 276 is not a sum of two whole-number squares — the prime factor 3 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √276 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 16 box, because 2² + 4² + 16² = 276.
• Since √276 = 2√69, a length of √276 is exactly 2 copies of the length √69 laid end to end.

Square roots near √276 and related roots

• The cube root of 276 is about 6.510830.
• Because 276 = 4 × 69, the root is twice √69: 2 × 8.306624 ≈ 16.613248.

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