Square Root of 825

√825 at a glance

Exact value

5√33

Decimal (10 places)

28.7228132327

Rounded

28.7 · 28.72 · 28.723

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.722813

Prime factorization

3 × 5² × 11

Cube root

9.378887

How to simplify √825

Look for the largest perfect square that divides 825. Here it is 25 (5²), because 825 = 25 × 33 and 33 has no square factor left:

The prime factorization tells the same story: 825 = 3 × 5² × 11. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 3 × 11 stays inside.

Check: (5√33)² = 5² × 33 = 25 × 33 = 825. As a decimal, 5√33 = 5 × 5.7445626465 ≈ 28.7228132327.

Where √825 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √825 lies between 28 and 29. 825 is 41 above 784 and 16 below 841, so the root is closer to 29.

• Straight line between 784 and 841: 28.7193 (0.01% low)
• Tangent from 28, i.e. 28 + 41 ÷ 56: 28.7321 (0.03% high)
• Tangent from 29, i.e. 29 − 16 ÷ 58: 28.7241 (0% high)

For √825 the tangent at 29 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 825 is just 16 below 841.

Finding √825 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 825: following the tangent line down to zero simplifies to averaging x with 825 ÷ x.

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 √825 = 28.7228132327 to every decimal shown.

√825 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √825 the pattern is [28; 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √825 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 − 825y² = 1. Its smallest solution in positive whole numbers is x = 48,599, y = 1,692.

√825 in geometry and everyday measurements

• 825 square feet is 76.6 m². Laid out as a square — a small house footprint or a lot — it is about 28.72 ft (28 ft 9 in) on a side.
• 825 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √825 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 14 × 25 box, because 2² + 14² + 25² = 825.
• Since √825 = 5√33, a length of √825 is exactly 5 copies of the length √33 laid end to end.

Square roots near √825 and related roots

• The cube root of 825 is about 9.378887.
• Squaring undoes the root: (√825)² = 825, while 825² = 680,625 — the number whose square root is 825.

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