Square Root 1 to 25 — Values, Chart & Examples

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.

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.