Square Root of 804

√804 at a glance

Exact value

2√201

Decimal (10 places)

28.3548937575

Rounded

28.4 · 28.35 · 28.355

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.354894

Prime factorization

2² × 3 × 67

Cube root

9.298624

How to simplify √804

Look for the largest perfect square that divides 804. Here it is 4 (2²), because 804 = 4 × 201 and 201 has no square factor left:

The prime factorization tells the same story: 804 = 2² × 3 × 67. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 67 stays inside.

Check: (2√201)² = 2² × 201 = 4 × 201 = 804. As a decimal, 2√201 = 2 × 14.1774468788 ≈ 28.3548937575.

Where √804 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √804 lies between 28 and 29. 804 is 20 above 784 and 37 below 841, so the root is closer to 28.

• Straight line between 784 and 841: 28.3509 (0.01% low)
• Tangent from 28, i.e. 28 + 20 ÷ 56: 28.3571 (0.01% high)
• Tangent from 29, i.e. 29 − 37 ÷ 58: 28.3621 (0.03% high)

For √804 the tangent at 28 wins, missing by only 0.0022. Tangent estimates shine when the number sits close to a perfect square — here 804 is just 20 above 784.

Finding √804 with the Babylonian method

Picture a rectangle with an area of 804 and one side x; the other side must be 804 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √804.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √804 = 28.3548937575 to every decimal shown.

√804 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √804 the pattern is [28; 2, 1, 4, 2, 18, 2, 4, 1, 2, 56] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √804 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 − 804y² = 1. Its smallest solution in positive whole numbers is x = 515,095, y = 18,166.

√804 in geometry and everyday measurements

• 804 square feet is 74.7 m². Laid out as a square — a small house footprint or a lot — it is about 28.35 ft (28 ft 4 in) on a side.
• 804 is not a sum of two whole-number squares — the prime factor 3 and 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √804 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 28 box, because 2² + 4² + 28² = 804.
• Since √804 = 2√201, a length of √804 is exactly 2 copies of the length √201 laid end to end.

Square roots near √804 and related roots

• The cube root of 804 is about 9.298624.
• Because 804 = 4 × 201, the root is twice √201: 2 × 14.177447 ≈ 28.354894.

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