Square Root of 780

√780 at a glance

Exact value

2√195

Decimal (10 places)

27.9284800875

Rounded

27.9 · 27.93 · 27.928

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.928480

Prime factorization

2² × 3 × 5 × 13

Cube root

9.205164

How to simplify √780

Look for the largest perfect square that divides 780. Here it is 4 (2²), because 780 = 4 × 195 and 195 has no square factor left:

The prime factorization tells the same story: 780 = 2² × 3 × 5 × 13. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 5 × 13 stays inside.

Check: (2√195)² = 2² × 195 = 4 × 195 = 780. As a decimal, 2√195 = 2 × 13.9642400438 ≈ 27.9284800875.

Where √780 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √780 lies between 27 and 28. 780 is 51 above 729 and 4 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.9273 (0% low)
• Tangent from 27, i.e. 27 + 51 ÷ 54: 27.9444 (0.06% high)
• Tangent from 28, i.e. 28 − 4 ÷ 56: 27.9286 (0% high)

For √780 the tangent at 28 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 780 is just 4 below 784.

Finding √780 with the Babylonian method

Picture a rectangle with an area of 780 and one side x; the other side must be 780 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √780.

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

The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √780 = 27.9284800875 to every decimal shown.

√780 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √780 the pattern is [27; 1, 12, 1, 54] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √780 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 − 780y² = 1. Its smallest solution in positive whole numbers is x = 391, y = 14.

√780 in geometry and everyday measurements

• 780 square feet is 72.5 m². Laid out as a square — a small house footprint or a lot — it is about 27.93 ft (27 ft 11 in) on a side.
• 780 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √780 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 10 × 26 box, because 2² + 10² + 26² = 780.
• Since √780 = 2√195, a length of √780 is exactly 2 copies of the length √195 laid end to end.

Square roots near √780 and related roots

• The cube root of 780 is about 9.205164.
• Because 780 = 4 × 195, the root is twice √195: 2 × 13.96424 ≈ 27.92848.

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