Square Root of 816

√816 at a glance

Exact value

4√51

Decimal (10 places)

28.5657137142

Rounded

28.6 · 28.57 · 28.566

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.565714

Prime factorization

2⁴ × 3 × 17

Cube root

9.344657

How to simplify √816

Look for the largest perfect square that divides 816. Here it is 16 (4²), because 816 = 16 × 51 and 51 has no square factor left:

The prime factorization tells the same story: 816 = 2⁴ × 3 × 17. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 3 × 17 stays inside.

816 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √816 = 2√204, and √204 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√51)² = 4² × 51 = 16 × 51 = 816. As a decimal, 4√51 = 4 × 7.1414284285 ≈ 28.5657137142.

Where √816 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √816 lies between 28 and 29. 816 is 32 above 784 and 25 below 841, so the root is closer to 29.

• Straight line between 784 and 841: 28.5614 (0.02% low)
• Tangent from 28, i.e. 28 + 32 ÷ 56: 28.5714 (0.02% high)
• Tangent from 29, i.e. 29 − 25 ÷ 58: 28.5690 (0.01% high)

For √816 the tangent at 29 wins, missing by only 0.0033. Tangent estimates shine when the number sits close to a perfect square — here 816 is just 25 below 841.

Finding √816 with the Babylonian method

Picture a rectangle with an area of 816 and one side x; the other side must be 816 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √816.

Start from the nearest whole number, 29 (29² = 841):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √816 = 28.5657137142 to every decimal shown.

√816 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √816 the pattern is [28; 1, 1, 3, 3, 3, 1, 1, 56] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √816 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 − 816y² = 1. Its smallest solution in positive whole numbers is x = 4,999, y = 175.

√816 in geometry and everyday measurements

• 816 square feet is 75.8 m². Laid out as a square — a small house footprint or a lot — it is about 28.57 ft (28 ft 7 in) on a side.
• 816 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 √816 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 4 × 28 box, because 4² + 4² + 28² = 816.
• Since √816 = 4√51, a length of √816 is exactly 4 copies of the length √51 laid end to end.

Square roots near √816 and related roots

• The cube root of 816 is about 9.344657.
• Because 816 = 4 × 204, the root is twice √204: 2 × 14.282857 ≈ 28.565714.

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