Square Root of 840

√840 at a glance

Exact value

2√210

Decimal (10 places)

28.9827534924

Rounded

29.0 · 28.98 · 28.983

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.982753

Prime factorization

2³ × 3 × 5 × 7

Cube root

9.435388

How to simplify √840

Look for the largest perfect square that divides 840. Here it is 4 (2²), because 840 = 4 × 210 and 210 has no square factor left:

The prime factorization tells the same story: 840 = 2³ × 3 × 5 × 7. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 × 5 × 7 stays inside.

Check: (2√210)² = 2² × 210 = 4 × 210 = 840. As a decimal, 2√210 = 2 × 14.4913767462 ≈ 28.9827534924.

Where √840 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √840 lies between 28 and 29. 840 is 56 above 784 and 1 below 841, so the root is closer to 29.

• Straight line between 784 and 841: 28.9825 (0% low)
• Tangent from 28, i.e. 28 + 56 ÷ 56: 29.0000 (0.06% high)
• Tangent from 29, i.e. 29 − 1 ÷ 58: 28.9828 (0% high)

For √840 the tangent at 29 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 840 is just 1 below 841.

Finding √840 with the Babylonian method

Picture a rectangle with an area of 840 and one side x; the other side must be 840 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √840.

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

Because the starting guess was already close, two steps are enough to match √840 = 28.9827534924 to every decimal shown.

√840 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √840 the pattern is [28; 1, 56] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √840 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 − 840y² = 1. Its smallest solution in positive whole numbers is x = 29, y = 1.

√840 in geometry and everyday measurements

• 840 square feet is 78 m². Laid out as a square — a small house footprint or a lot — it is about 28.98 ft (29 ft) on a side.
• 840 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √840 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 8 × 10 × 26 box, because 8² + 10² + 26² = 840.
• Since √840 = 2√210, a length of √840 is exactly 2 copies of the length √210 laid end to end.

Square roots near √840 and related roots

• The cube root of 840 is about 9.435388.
• Because 840 = 4 × 210, the root is twice √210: 2 × 14.491377 ≈ 28.982753.

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