Square Root 1 to 25 Chart
The value of the square root of a number $n$ is the number that, multiplied by itself, gives $n$. From 1 to 25 only five inputs land on a clean whole number; the rest are non-terminating decimals. Here is the complete chart, with perfect squares shown exactly and every other value rounded to three decimal places.
Number | Square Root | Number | Square Root | Number | Square Root |
|---|---|---|---|---|---|
$\sqrt{1}$ | $1$ (exact) | $\sqrt{10}$ | $3.162$ | $\sqrt{19}$ | $4.359$ |
$\sqrt{2}$ | $1.414$ | $\sqrt{11}$ | $3.317$ | $\sqrt{20}$ | $4.472$ |
$\sqrt{3}$ | $1.732$ | $\sqrt{12}$ | $3.464$ | $\sqrt{21}$ | $4.583$ |
$\sqrt{4}$ | $2$ (exact) | $\sqrt{13}$ | $3.606$ | $\sqrt{22}$ | $4.690$ |
$\sqrt{5}$ | $2.236$ | $\sqrt{14}$ | $3.742$ | $\sqrt{23}$ | $4.796$ |
$\sqrt{6}$ | $2.449$ | $\sqrt{15}$ | $3.873$ | $\sqrt{24}$ | $4.899$ |
$\sqrt{7}$ | $2.646$ | $\sqrt{16}$ | $4$ (exact) | $\sqrt{25}$ | $5$ (exact) |
$\sqrt{8}$ | $2.828$ | $\sqrt{17}$ | $4.123$ | ||
$\sqrt{9}$ | $3$ (exact) | $\sqrt{18}$ | $4.243$ |
Which Square Roots From 1 to 25 Are Rational
A rational number can be written as a fraction of two integers; an irrational number cannot, and its decimal never terminates or repeats. Only the perfect squares in this range give rational roots.
Rational (perfect squares): $\sqrt{1}, \sqrt{4}, \sqrt{9}, \sqrt{16}, \sqrt{25}$ (five values).
Irrational (everything else): $\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7}, \sqrt{8}, \sqrt{10}, \sqrt{11}, \sqrt{12}, \sqrt{13}, \sqrt{14}, \sqrt{15}, \sqrt{17}, \sqrt{18}, \sqrt{19}, \sqrt{20}, \sqrt{21}, \sqrt{22}, \sqrt{23}, \sqrt{24}$ (twenty values).
So exactly $\tfrac{5}{25}$, or one in five, of the roots from 1 to 25 are rational. That fraction is worth internalising: perfect squares are rare, and the gaps between them grow as numbers climb.
Where Square Roots From 1 to 25 Appear
These small square roots show up constantly. $\sqrt{2} \approx 1.414$ is the diagonal of a unit square, so it lives in every set square and every 45° cut a carpenter makes. $\sqrt{5} \approx 2.236$ is the diagonal of a $1 \times 2$ rectangle, and it hides inside the golden ratio $\tfrac{1+\sqrt{5}}{2}$. Screen and paper sizes, standard-deviation calculations, and the distance formula in coordinate geometry all lean on roots in this range.
How to Read and Use the Table
Read each entry as a question and an answer: $\sqrt{7} = 2.646$ means "the number whose square is 7 is about 2.646." Check it and $2.646^2 = 7.001$, close enough for the three-decimal rounding.
To use the chart for estimation, locate your number between two perfect squares. For $\sqrt{7}$, note that $4 < 7 < 9$, so the answer sits between $\sqrt{4}=2$ and $\sqrt{9}=3$. Because 7 is closer to 9, the root leans toward 3, which matches 2.646.
The point of the chart is not to memorise twenty decimals. It is to see the structure: exact roots at 1, 4, 9, 16, 25, and a predictable climb in between that you can reconstruct with the perfect-square anchors on either side.
How to Compute Square Roots From 1 to 25
Method 1: Perfect-square recognition
For $\sqrt{16}$, ask which whole number times itself is 16. $4 \times 4 = 16$ So $\sqrt{16} = 4$.
The same works for 1, 4, 9, and 25. No decimals needed.
Method 2: Estimation between anchors (for non-perfect squares)
Take $\sqrt{12}$. Find the nearest perfect squares below and above: $9$ and $16$. So $3 < \sqrt{12} < 4$. Since $12$ is closer to $9$ than to $16$, estimate near $3.4$. Refine: $3.4^2 = 11.56$ and $3.5^2 = 12.25$, so the root is between them, near $3.46$. Final answer: $\sqrt{12} \approx 3.464$.
Method 3: Long division (for a precise decimal)
Take $\sqrt{20}$. Pair digits from the decimal point: $20.\overline{00}\,\overline{00}$. Largest square $\le 20$ is $16 = 4^2$, so the first digit is $4$, remainder $4$. Bring down $00$ to get $400$; double the quotient (4 → 8) and find a digit $d$ with $8d \times d \le 400$; $84 \times 4 = 336$, so next digit is $4$, remainder $64$. Continue to reach $4.472$. Final answer: $\sqrt{20} \approx 4.472$.
Common Mistakes With Square Root 1 to 25
Mistake 1: Treating the decimal as exact
Where it slips in: Writing $\sqrt{2} = 1.414$ with an equals sign in an exact answer.
Don't do this: State $\sqrt{2} = 1.414$ as if the decimal ends.
The correct way: Keep the radical for exact work, $\sqrt{2}$, and use $\approx 1.414$ only when a decimal is asked for.
Mistake 2: Confusing squares with square roots
Where it slips in: Reading a "1 to 25" chart and mixing up $\sqrt{25} = 5$ with $25^2 = 625$.
Don't do this: Report $\sqrt{9} = 81$.
The correct way: The square root shrinks the number back down: $\sqrt{9} = 3$, because $3^2 = 9$.
Mistake 3: Assuming every root simplifies to a fraction
Where it slips in: Trying to write $\sqrt{7}$ as a neat fraction.
Don't do this: Claim $\sqrt{7} = \tfrac{26}{10}$ because $2.6$ looks close.
The correct way: $\sqrt{7}$ is irrational; no fraction equals it exactly. The one habit that fixes this: check whether the number is a perfect square first, and if it is not, the root is irrational.
Conclusion
The square root 1 to 25 chart holds five exact values ($\sqrt{1}, \sqrt{4}, \sqrt{9}, \sqrt{16}, \sqrt{25}$) and twenty irrational ones.
Rational roots come only from perfect squares; the other twenty are non-terminating decimals kept in radical form for exact work.
Estimate any non-perfect square by trapping it between the nearest perfect-square anchors.
Read the table as structure, not as decimals to memorise.
To build this table understanding with a teacher, explore Bhanzu's algebra tutor, get help with algebra, or browse math classes online.
Read More
Square Root of 10 — the diagonal-of-a-rectangle root explained in full.
Square Root of 115 — a larger irrational root and its simplest form.
Squares and Square Roots — the concept behind the chart.
What Is a Square Root — the definition and notation from scratch.
Square Root of 25 — a closer look at the largest exact root in this range.
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