Square Root of 824

√824 at a glance

Exact value

2√206

Decimal (10 places)

28.7054001888

Rounded

28.7 · 28.71 · 28.705

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.705400

Prime factorization

2³ × 103

Cube root

9.375096

How to simplify √824

Look for the largest perfect square that divides 824. Here it is 4 (2²), because 824 = 4 × 206 and 206 has no square factor left:

The prime factorization tells the same story: 824 = 2³ × 103. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 103 stays inside.

Check: (2√206)² = 2² × 206 = 4 × 206 = 824. As a decimal, 2√206 = 2 × 14.3527000944 ≈ 28.7054001888.

Where √824 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √824 lies between 28 and 29. 824 is 40 above 784 and 17 below 841, so the root is closer to 29.

• Straight line between 784 and 841: 28.7018 (0.01% low)
• Tangent from 28, i.e. 28 + 40 ÷ 56: 28.7143 (0.03% high)
• Tangent from 29, i.e. 29 − 17 ÷ 58: 28.7069 (0.01% high)

For √824 the tangent at 29 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 824 is just 17 below 841.

Finding √824 with the Babylonian method

Picture a rectangle with an area of 824 and one side x; the other side must be 824 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √824.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √824 = 28.7054001888 to every decimal shown.

√824 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √824 the pattern is [28; 1, 2, 2, 1, 1, 6, 1, 1, 2, 2, 1, 56] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √824 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 − 824y² = 1. Its smallest solution in positive whole numbers is x = 59,535, y = 2,074.

√824 in geometry and everyday measurements

• 824 square feet is 76.6 m². Laid out as a square — a small house footprint or a lot — it is about 28.71 ft (28 ft 8 in) on a side.
• 824 is not a sum of two whole-number squares — the prime factor 103 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √824 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 28 box, because 2² + 6² + 28² = 824.
• Since √824 = 2√206, a length of √824 is exactly 2 copies of the length √206 laid end to end.

Square roots near √824 and related roots

• The cube root of 824 is about 9.375096.
• Because 824 = 4 × 206, the root is twice √206: 2 × 14.3527 ≈ 28.7054.

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