The square root of 360 is approximately 18.974, and its exact simplified form is $6\sqrt{10}$.
Quick Answer:
Result: $\sqrt{360} = 6\sqrt{10} \approx 18.974$
Notation: Simplified radical $6\sqrt{10}$; decimal $\approx 18.974$ (irrational, non-terminating)
Method shown: Prime factorization to simplify the radical, plus estimation between perfect squares
Approximate value: $18.9736659...$
Exact form: $6\sqrt{10}$ (never a whole number, since 360 is not a perfect square)
Quick Reference Table
Number $n$ | $\sqrt{n}$ exact | $\sqrt{n}$ approx |
|---|---|---|
324 | $18$ | $18.000$ |
360 | $6\sqrt{10}$ | $18.974$ |
361 | $19$ | $19.000$ |
400 | $20$ | $20.000$ |
90 | $3\sqrt{10}$ | $9.487$ |
1000 | $10\sqrt{10}$ | $31.623$ |
10 | $\sqrt{10}$ | $3.162$ |
Every $\sqrt{10}$ row is a multiple of $3.162$: $\sqrt{360} = 6\sqrt{10}$ is exactly six times $\sqrt{10}$.
Where the Square Root of 360 Appears
A circle has 360 degrees, so $\sqrt{360}$ turns up when a quantity scales with the square root of a full rotation. It also appears as the diagonal length in geometry problems: a rectangle with area 360 and sides in the ratio $6:10$ has a diagonal tied to this value. Any geometric-mean calculation whose two numbers multiply to 360 lands on $\sqrt{360}$.
What a Square Root Means Here
A square root of a number is the value that, multiplied by itself, gives that number. No whole number squared equals 360 ($18^2 = 324$ and $19^2 = 361$), so the square root of 360 is irrational — its decimal never ends or repeats. We keep the exact form $6\sqrt{10}$ next to the rounded decimal.
How to Find the Square Root of 360
Method 1: Prime factorization (gives the exact simplified form)
Break 360 into primes.
$360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5$
Group identical factors into pairs.
$360 = (2 \times 2) \times (3 \times 3) \times (2 \times 5)$
Each complete pair leaves the radical as a single factor.
$\sqrt{360} = 2 \times 3 \times \sqrt{2 \times 5}$
$\sqrt{360} = 6\sqrt{10}$
Final answer: $6\sqrt{10}$
Method 2: Largest perfect-square factor (a faster shortcut)
The biggest perfect square dividing 360 is 36.
$360 = 36 \times 10$
Split the radical across the product.
$\sqrt{360} = \sqrt{36} \times \sqrt{10}$
$\sqrt{360} = 6\sqrt{10}$
Final answer: $6\sqrt{10}$
Method 3: Estimating the decimal by hand
Place 360 between two perfect squares.
$18^2 = 324 \quad \text{and} \quad 19^2 = 361$
So the root is just under 19, since 360 is almost 361.
Test 18.97.
$18.97^2 = 359.86$
Test 18.98.
$18.98^2 = 360.24$
The root sits between 18.97 and 18.98, giving $\sqrt{360} \approx 18.974$.
Final answer: $\approx 18.974$
Common Mistakes With Square Root of 360
Mistake 1: Choosing a smaller perfect-square factor and stopping
Where it slips in: Splitting 360 as $4 \times 90$ and writing $2\sqrt{90}$ as final.
Don't do this: Leaving $2\sqrt{90}$, because 90 still contains the perfect square 9.
The correct way: Use the largest perfect square (36), or keep simplifying $2\sqrt{90} = 2 \times 3\sqrt{10} = 6\sqrt{10}$.
Mistake 2: Multiplying the outside and inside numbers together
Where it slips in: Reading $6\sqrt{10}$ and computing $60$.
Don't do this: Writing $6\sqrt{10} = \sqrt{60}$ or $= 60$; the 6 is a coefficient, not part of the radicand.
The correct way: $6\sqrt{10}$ means $6 \times \sqrt{10} = 6 \times 3.162 \approx 18.974$.
Mistake 3: Assuming 360 is a perfect square
Where it slips in: Expecting a clean whole-number answer because 360 is a familiar number.
The correct way: Check the nearest squares first ($18^2 = 324$, $19^2 = 361$); since 360 is between them, the root is irrational.
Conclusion
The square root of 360 is $6\sqrt{10}$ in exact form and about $18.974$ as a decimal.
Prime factorization ($2^3 \times 3^2 \times 5$) or the largest-perfect-square shortcut ($36 \times 10$) both give $6\sqrt{10}$.
Because 360 is not a perfect square, the root is irrational — keep the radical for exact answers.
To get comfortable simplifying radicals like these, work with an algebra tutor or explore help with algebra.
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