Estimating Square Roots with Perfect Squares

Property To approximate an irrational square root $\sqrt{n}$ to the nearest integer, find the two consecutive perfect squares that $n$ is between. If $a^2 < n < (a+1)^2$, then the value of $\sqrt{n}$ is strictly between the integers $a$ and $a+1$: $$a < \sqrt{n} < a+1$$.

Examples Approximate $\sqrt{30}$ to the nearest integer. The perfect squares closest to $30$ are $25$ and $36$. $$25 < 30 < 36$$ $$\sqrt{25} < \sqrt{30} < \sqrt{36}$$ $$5 < \sqrt{30} < 6$$ Since $30$ is closer to $25$ than to $36$, the best integer approximation is $5$. So, $\sqrt{30} \approx 5$.

Approximate $\sqrt{85}$ to the nearest integer. The perfect squares closest to $85$ are $81$ and $100$. $$81 < 85 < 100$$ $$\sqrt{81} < \sqrt{85} < \sqrt{100}$$ $$9 < \sqrt{85} < 10$$ Since $85$ is closer to $81$ than to $100$, the best integer approximation is $9$. So, $\sqrt{85} \approx 9$.