Square Root of 812

√812 at a glance

Exact value

2√203

Decimal (10 places)

28.4956136976

Rounded

28.5 · 28.50 · 28.496

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.495614

Prime factorization

2² × 7 × 29

Cube root

9.329363

How to simplify √812

Look for the largest perfect square that divides 812. Here it is 4 (2²), because 812 = 4 × 203 and 203 has no square factor left:

The prime factorization tells the same story: 812 = 2² × 7 × 29. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 7 × 29 stays inside.

Check: (2√203)² = 2² × 203 = 4 × 203 = 812. As a decimal, 2√203 = 2 × 14.2478068488 ≈ 28.4956136976.

Where √812 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √812 lies between 28 and 29. 812 is 28 above 784 and 29 below 841, so the root is closer to 28.

• Straight line between 784 and 841: 28.4912 (0.02% low)
• Tangent from 28, i.e. 28 + 28 ÷ 56: 28.5000 (0.02% high)
• Tangent from 29, i.e. 29 − 29 ÷ 58: 28.5000 (0.02% high)

For √812 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √812 with the Babylonian method

Picture a rectangle with an area of 812 and one side x; the other side must be 812 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √812.

Start from the nearest whole number, 28 (28² = 784):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √812 = 28.4956136976 to every decimal shown.

√812 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √812 the pattern is [28; 2, 56] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √812 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 − 812y² = 1. Its smallest solution in positive whole numbers is x = 57, y = 2.

√812 in geometry and everyday measurements

• 812 square feet is 75.4 m². Laid out as a square — a small house footprint or a lot — it is about 28.5 ft (28 ft 6 in) on a side.
• 812 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √812 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 18 × 22 box, because 2² + 18² + 22² = 812.
• Since √812 = 2√203, a length of √812 is exactly 2 copies of the length √203 laid end to end.

Square roots near √812 and related roots

• The cube root of 812 is about 9.329363.
• Because 812 = 4 × 203, the root is twice √203: 2 × 14.247807 ≈ 28.495614.

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